Properties

Label 4-112e2-1.1-c6e2-0-0
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $663.889$
Root an. cond. $5.07602$
Motivic weight $6$
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  − 266·7-s − 582·9-s − 1.74e3·11-s − 9.47e3·23-s + 2.92e4·25-s + 2.22e4·29-s + 6.00e3·37-s − 6.28e4·43-s − 4.68e4·49-s − 1.52e5·53-s + 1.54e5·63-s − 9.90e5·67-s + 3.68e5·71-s + 4.64e5·77-s + 1.06e6·79-s − 1.92e5·81-s + 1.01e6·99-s − 3.23e6·107-s + 3.98e5·109-s − 3.61e6·113-s − 1.25e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.775·7-s − 0.798·9-s − 1.31·11-s − 0.778·23-s + 1.86·25-s + 0.914·29-s + 0.118·37-s − 0.790·43-s − 0.398·49-s − 1.02·53-s + 0.619·63-s − 3.29·67-s + 1.03·71-s + 1.01·77-s + 2.16·79-s − 0.362·81-s + 1.04·99-s − 2.63·107-s + 0.307·109-s − 2.50·113-s − 0.706·121-s + 0.603·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(663.889\)
Root analytic conductor: \(5.07602\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12544,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.5163562114\)
\(L(\frac12)\) \(\approx\) \(0.5163562114\)
\(L(4)\) 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 \( 1 \)
7$C_2$ \( 1 + 38 p T + p^{6} T^{2} \)
good3$C_2^2$ \( 1 + 194 p T^{2} + p^{12} T^{4} \)
5$C_2^2$ \( 1 - 5842 p T^{2} + p^{12} T^{4} \)
11$C_2$ \( ( 1 + 874 T + p^{6} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 4755578 T^{2} + p^{12} T^{4} \)
17$C_2^2$ \( 1 - 748834 p T^{2} + p^{12} T^{4} \)
19$C_2^2$ \( 1 - 84379322 T^{2} + p^{12} T^{4} \)
23$C_2$ \( ( 1 + 206 p T + p^{6} T^{2} )^{2} \)
29$C_2$ \( ( 1 - 11146 T + p^{6} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 1020892802 T^{2} + p^{12} T^{4} \)
37$C_2$ \( ( 1 - 3002 T + p^{6} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6189133442 T^{2} + p^{12} T^{4} \)
43$C_2$ \( ( 1 + 31418 T + p^{6} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 16309886018 T^{2} + p^{12} T^{4} \)
53$C_2$ \( ( 1 + 76406 T + p^{6} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 71539567322 T^{2} + p^{12} T^{4} \)
61$C_2^2$ \( 1 - 27356175482 T^{2} + p^{12} T^{4} \)
67$C_2$ \( ( 1 + 495242 T + p^{6} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 184406 T + p^{6} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 298950552578 T^{2} + p^{12} T^{4} \)
79$C_2$ \( ( 1 - 534934 T + p^{6} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 142873131578 T^{2} + p^{12} T^{4} \)
89$C_2^2$ \( 1 - 597656180162 T^{2} + p^{12} T^{4} \)
97$C_2^2$ \( 1 - 1002631840898 T^{2} + p^{12} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.13993357787200253256655506065, −12.00907090358046090597276009343, −11.99264056979823282148928434181, −10.94713503559953690667192637801, −10.62226182277916047184674247722, −10.24517907146753832095005386178, −9.438799918308330870951007759284, −9.072811271749236617184909214262, −8.162791081241336266117612346859, −8.092747967334230954732713323839, −7.16141105188838870396674138524, −6.49377077789719751102827339783, −6.06413230432099244801668572873, −5.18053809759035456296527067338, −4.82689185657972481967586583733, −3.75856884942224993675502341503, −2.84915580985799963762364128930, −2.68195708644248051985688528437, −1.36880019733931573511789669625, −0.24021167838274501703095140520, 0.24021167838274501703095140520, 1.36880019733931573511789669625, 2.68195708644248051985688528437, 2.84915580985799963762364128930, 3.75856884942224993675502341503, 4.82689185657972481967586583733, 5.18053809759035456296527067338, 6.06413230432099244801668572873, 6.49377077789719751102827339783, 7.16141105188838870396674138524, 8.092747967334230954732713323839, 8.162791081241336266117612346859, 9.072811271749236617184909214262, 9.438799918308330870951007759284, 10.24517907146753832095005386178, 10.62226182277916047184674247722, 10.94713503559953690667192637801, 11.99264056979823282148928434181, 12.00907090358046090597276009343, 13.13993357787200253256655506065

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