Properties

Label 24-336e12-1.1-c3e12-0-1
Degree $24$
Conductor $2.070\times 10^{30}$
Sign $1$
Analytic cond. $3.68522\times 10^{15}$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 38·9-s − 96·13-s + 216·25-s − 720·37-s − 294·49-s + 432·61-s + 1.65e3·73-s + 505·81-s − 6.26e3·97-s + 1.03e3·109-s − 3.64e3·117-s − 3.02e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7.90e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 1.40·9-s − 2.04·13-s + 1.72·25-s − 3.19·37-s − 6/7·49-s + 0.906·61-s + 2.65·73-s + 0.692·81-s − 6.55·97-s + 0.906·109-s − 2.88·117-s − 2.27·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 3.59·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(3.68522\times 10^{15}\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{336} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 3^{12} \cdot 7^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(3.554840740\)
\(L(\frac12)\) \(\approx\) \(3.554840740\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 38 T^{2} + 313 p T^{4} - 1148 p^{2} T^{6} + 313 p^{7} T^{8} - 38 p^{12} T^{10} + p^{18} T^{12} \)
7 \( ( 1 + p^{2} T^{2} )^{6} \)
good5 \( ( 1 - 108 T^{2} + 1431 p T^{4} - 1796312 T^{6} + 1431 p^{7} T^{8} - 108 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
11 \( ( 1 + 1512 T^{2} + 3234063 T^{4} + 2362820192 T^{6} + 3234063 p^{6} T^{8} + 1512 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
13 \( ( 1 + 24 T + 3417 T^{2} + 27172 T^{3} + 3417 p^{3} T^{4} + 24 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
17 \( ( 1 - 14844 T^{2} + 131288487 T^{4} - 785980991096 T^{6} + 131288487 p^{6} T^{8} - 14844 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
19 \( ( 1 - 26586 T^{2} + 357012507 T^{4} - 3019276931428 T^{6} + 357012507 p^{6} T^{8} - 26586 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
23 \( ( 1 + 44616 T^{2} + 1065610983 T^{4} + 30205429568 p^{2} T^{6} + 1065610983 p^{6} T^{8} + 44616 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
29 \( ( 1 - 108270 T^{2} + 5419906887 T^{4} - 164281853714372 T^{6} + 5419906887 p^{6} T^{8} - 108270 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
31 \( ( 1 - 158646 T^{2} + 11015478687 T^{4} - 427590660466388 T^{6} + 11015478687 p^{6} T^{8} - 158646 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
37 \( ( 1 + 180 T + 108651 T^{2} + 15672664 T^{3} + 108651 p^{3} T^{4} + 180 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
41 \( ( 1 - 4716 p T^{2} + 22580988663 T^{4} - 1745347169586968 T^{6} + 22580988663 p^{6} T^{8} - 4716 p^{13} T^{10} + p^{18} T^{12} )^{2} \)
43 \( ( 1 - 157146 T^{2} + 22424682903 T^{4} - 1999243853829676 T^{6} + 22424682903 p^{6} T^{8} - 157146 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
47 \( ( 1 + 309714 T^{2} + 58339927119 T^{4} + 7161872589826684 T^{6} + 58339927119 p^{6} T^{8} + 309714 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
53 \( ( 1 - 465654 T^{2} + 132219070935 T^{4} - 23481979333640756 T^{6} + 132219070935 p^{6} T^{8} - 465654 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
59 \( ( 1 + 759810 T^{2} + 245204587515 T^{4} + 53947255679253236 T^{6} + 245204587515 p^{6} T^{8} + 759810 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
61 \( ( 1 - 108 T + 294513 T^{2} - 39131028 T^{3} + 294513 p^{3} T^{4} - 108 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
67 \( ( 1 - 742458 T^{2} + 337103376231 T^{4} - 119454889028429612 T^{6} + 337103376231 p^{6} T^{8} - 742458 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
71 \( ( 1 + 1517904 T^{2} + 1045389452967 T^{4} + 450915097427172560 T^{6} + 1045389452967 p^{6} T^{8} + 1517904 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
73 \( ( 1 - 414 T + 280767 T^{2} - 44414452 T^{3} + 280767 p^{3} T^{4} - 414 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
79 \( ( 1 - 1224426 T^{2} + 1012954623663 T^{4} - 545208850365051404 T^{6} + 1012954623663 p^{6} T^{8} - 1224426 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
83 \( ( 1 + 528810 T^{2} + 429704724651 T^{4} + 394687164189819332 T^{6} + 429704724651 p^{6} T^{8} + 528810 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
89 \( ( 1 - 3607020 T^{2} + 5707274044311 T^{4} - 5162372304755975768 T^{6} + 5707274044311 p^{6} T^{8} - 3607020 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
97 \( ( 1 + 1566 T + 3501615 T^{2} + 2975935204 T^{3} + 3501615 p^{3} T^{4} + 1566 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.41206125758669668144873807571, −3.23348519471406350390522244835, −3.04101459571552117070506070340, −2.96067673348780749747650606310, −2.94875801268438841338068734913, −2.94749932909651930605157095015, −2.70449369670599026849114737275, −2.50911486267979513643280238655, −2.42323522439314134773580588061, −2.39893836580177432338743985545, −2.15077785817859108566317828305, −2.10015595523158780743665878514, −2.03596505910109140477091251367, −1.70996419128607998155574944977, −1.54788332127628642474475857619, −1.52213892353533966083238584491, −1.49141872216298845711340809048, −1.37453238179389073134509413388, −1.08602230662034672769811064608, −0.928143942797596456591230828070, −0.74649762962773082023561629780, −0.62652500995617171403182603631, −0.34547317786873830648959549179, −0.27293341319712162881370459328, −0.13300623891011435106979809880, 0.13300623891011435106979809880, 0.27293341319712162881370459328, 0.34547317786873830648959549179, 0.62652500995617171403182603631, 0.74649762962773082023561629780, 0.928143942797596456591230828070, 1.08602230662034672769811064608, 1.37453238179389073134509413388, 1.49141872216298845711340809048, 1.52213892353533966083238584491, 1.54788332127628642474475857619, 1.70996419128607998155574944977, 2.03596505910109140477091251367, 2.10015595523158780743665878514, 2.15077785817859108566317828305, 2.39893836580177432338743985545, 2.42323522439314134773580588061, 2.50911486267979513643280238655, 2.70449369670599026849114737275, 2.94749932909651930605157095015, 2.94875801268438841338068734913, 2.96067673348780749747650606310, 3.04101459571552117070506070340, 3.23348519471406350390522244835, 3.41206125758669668144873807571

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.