Properties

Label 2-350-1.1-c7-0-47
Degree $2$
Conductor $350$
Sign $1$
Analytic cond. $109.334$
Root an. cond. $10.4563$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 82·3-s + 64·4-s + 656·6-s + 343·7-s + 512·8-s + 4.53e3·9-s + 2.40e3·11-s + 5.24e3·12-s − 7.11e3·13-s + 2.74e3·14-s + 4.09e3·16-s − 2.48e3·17-s + 3.62e4·18-s + 3.64e4·19-s + 2.81e4·21-s + 1.92e4·22-s + 1.28e4·23-s + 4.19e4·24-s − 5.69e4·26-s + 1.92e5·27-s + 2.19e4·28-s − 8.80e4·29-s + 2.82e5·31-s + 3.27e4·32-s + 1.97e5·33-s − 1.98e4·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.75·3-s + 1/2·4-s + 1.23·6-s + 0.377·7-s + 0.353·8-s + 2.07·9-s + 0.545·11-s + 0.876·12-s − 0.898·13-s + 0.267·14-s + 1/4·16-s − 0.122·17-s + 1.46·18-s + 1.22·19-s + 0.662·21-s + 0.385·22-s + 0.220·23-s + 0.619·24-s − 0.635·26-s + 1.88·27-s + 0.188·28-s − 0.670·29-s + 1.70·31-s + 0.176·32-s + 0.956·33-s − 0.0867·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(109.334\)
Root analytic conductor: \(10.4563\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(8.102557407\)
\(L(\frac12)\) \(\approx\) \(8.102557407\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{3} T \)
5 \( 1 \)
7 \( 1 - p^{3} T \)
good3 \( 1 - 82 T + p^{7} T^{2} \)
11 \( 1 - 2408 T + p^{7} T^{2} \)
13 \( 1 + 7116 T + p^{7} T^{2} \)
17 \( 1 + 2486 T + p^{7} T^{2} \)
19 \( 1 - 36482 T + p^{7} T^{2} \)
23 \( 1 - 560 p T + p^{7} T^{2} \)
29 \( 1 + 88094 T + p^{7} T^{2} \)
31 \( 1 - 282636 T + p^{7} T^{2} \)
37 \( 1 - 214534 T + p^{7} T^{2} \)
41 \( 1 + 140874 T + p^{7} T^{2} \)
43 \( 1 + 848 p T + p^{7} T^{2} \)
47 \( 1 + 716868 T + p^{7} T^{2} \)
53 \( 1 - 56946 T + p^{7} T^{2} \)
59 \( 1 + 2149862 T + p^{7} T^{2} \)
61 \( 1 - 3084360 T + p^{7} T^{2} \)
67 \( 1 - 3034364 T + p^{7} T^{2} \)
71 \( 1 + 106624 T + p^{7} T^{2} \)
73 \( 1 + 988930 T + p^{7} T^{2} \)
79 \( 1 - 3415896 T + p^{7} T^{2} \)
83 \( 1 - 15142 T + p^{7} T^{2} \)
89 \( 1 - 174810 T + p^{7} T^{2} \)
97 \( 1 + 13506790 T + p^{7} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.982239273512018540142887522468, −9.393123106566862568910299299278, −8.283639899253426181926521537615, −7.58032331401675064331509092974, −6.67900794450008966774296401055, −5.10117934638576850928606136724, −4.13285164499215782834597883342, −3.15415952750196815640144555132, −2.33888454685084093184376309542, −1.24322877861927529392086457022, 1.24322877861927529392086457022, 2.33888454685084093184376309542, 3.15415952750196815640144555132, 4.13285164499215782834597883342, 5.10117934638576850928606136724, 6.67900794450008966774296401055, 7.58032331401675064331509092974, 8.283639899253426181926521537615, 9.393123106566862568910299299278, 9.982239273512018540142887522468

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