Properties

Label 4-768e2-1.1-c3e2-0-19
Degree $4$
Conductor $589824$
Sign $1$
Analytic cond. $2053.31$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 9·9-s − 60·11-s + 144·13-s − 288·23-s + 218·25-s + 108·27-s + 360·33-s + 144·37-s − 864·39-s + 1.15e3·47-s + 398·49-s + 828·59-s − 1.00e3·61-s + 1.72e3·69-s − 1.44e3·71-s + 356·73-s − 1.30e3·75-s − 891·81-s + 876·83-s + 1.30e3·97-s − 540·99-s + 3.32e3·107-s + 2.44e3·109-s − 864·111-s + 1.29e3·117-s + 38·121-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.64·11-s + 3.07·13-s − 2.61·23-s + 1.74·25-s + 0.769·27-s + 1.89·33-s + 0.639·37-s − 3.54·39-s + 3.57·47-s + 1.16·49-s + 1.82·59-s − 2.11·61-s + 3.01·69-s − 2.40·71-s + 0.570·73-s − 2.01·75-s − 1.22·81-s + 1.15·83-s + 1.36·97-s − 0.548·99-s + 3.00·107-s + 2.15·109-s − 0.738·111-s + 1.02·117-s + 0.0285·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(589824\)    =    \(2^{16} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(2053.31\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 589824,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.150312223\)
\(L(\frac12)\) \(\approx\) \(2.150312223\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 + 2 p T + p^{3} T^{2} \)
good5$C_2^2$ \( 1 - 218 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 398 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 30 T + p^{3} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 72 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 7234 T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 13070 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + 144 T + p^{3} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 48746 T^{2} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 10910 T^{2} + p^{6} T^{4} \)
37$C_2$ \( ( 1 - 72 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 44530 T^{2} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 106526 T^{2} + p^{6} T^{4} \)
47$C_2$ \( ( 1 - 576 T + p^{3} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 32762 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 414 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 504 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 21202 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 720 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 178 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 50366 T^{2} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 438 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 660850 T^{2} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 650 T + p^{3} T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.25725784727337164996811846893, −10.13905770089018809513913421169, −9.092027324973502993100444126076, −8.679600828303836860673818846089, −8.664461695791863888743106236926, −7.970453036261499286533284172918, −7.58074123748891396736717079958, −7.10276166727619576505529393788, −6.37192430143369178818245206847, −6.02191839819799156398782876341, −5.76460262935656581465090306527, −5.57743074664330087968787415132, −4.68995097896403613554006940970, −4.31992047217821206671967509175, −3.75168484433937866179783065936, −3.17363791106871008275545493131, −2.48556457673478477980894873681, −1.79349847616511650137310725010, −0.792591456520174013461032223982, −0.63775003804180060977512182512, 0.63775003804180060977512182512, 0.792591456520174013461032223982, 1.79349847616511650137310725010, 2.48556457673478477980894873681, 3.17363791106871008275545493131, 3.75168484433937866179783065936, 4.31992047217821206671967509175, 4.68995097896403613554006940970, 5.57743074664330087968787415132, 5.76460262935656581465090306527, 6.02191839819799156398782876341, 6.37192430143369178818245206847, 7.10276166727619576505529393788, 7.58074123748891396736717079958, 7.970453036261499286533284172918, 8.664461695791863888743106236926, 8.679600828303836860673818846089, 9.092027324973502993100444126076, 10.13905770089018809513913421169, 10.25725784727337164996811846893

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