Properties

Label 2-97461-1.1-c1-0-10
Degree $2$
Conductor $97461$
Sign $1$
Analytic cond. $778.230$
Root an. cond. $27.8967$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 4·5-s − 3·8-s − 4·10-s + 4·11-s − 13-s − 16-s + 17-s + 4·20-s + 4·22-s + 4·23-s + 11·25-s − 26-s + 4·29-s − 8·31-s + 5·32-s + 34-s + 4·37-s + 12·40-s − 4·43-s − 4·44-s + 4·46-s + 10·47-s + 11·50-s + 52-s − 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s − 1.26·10-s + 1.20·11-s − 0.277·13-s − 1/4·16-s + 0.242·17-s + 0.894·20-s + 0.852·22-s + 0.834·23-s + 11/5·25-s − 0.196·26-s + 0.742·29-s − 1.43·31-s + 0.883·32-s + 0.171·34-s + 0.657·37-s + 1.89·40-s − 0.609·43-s − 0.603·44-s + 0.589·46-s + 1.45·47-s + 1.55·50-s + 0.138·52-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97461\)    =    \(3^{2} \cdot 7^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(778.230\)
Root analytic conductor: \(27.8967\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{97461} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 97461,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.054142144\)
\(L(\frac12)\) \(\approx\) \(2.054142144\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 + T \)
17 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−14.05215177993410, −13.13408564456123, −12.61588373354552, −12.49645454087898, −11.86124296824900, −11.44587914747880, −11.15823144602248, −10.44735457105097, −9.715934065579616, −9.200612793709041, −8.751662393074139, −8.360342548413119, −7.676475694753683, −7.268662477854435, −6.675840068430440, −6.177329914151799, −5.375530041484257, −4.867611990696879, −4.372417632394640, −3.924406831459724, −3.414473880195038, −3.089525560646280, −2.093102010754489, −0.9475130866663913, −0.5264350175610795, 0.5264350175610795, 0.9475130866663913, 2.093102010754489, 3.089525560646280, 3.414473880195038, 3.924406831459724, 4.372417632394640, 4.867611990696879, 5.375530041484257, 6.177329914151799, 6.675840068430440, 7.268662477854435, 7.676475694753683, 8.360342548413119, 8.751662393074139, 9.200612793709041, 9.715934065579616, 10.44735457105097, 11.15823144602248, 11.44587914747880, 11.86124296824900, 12.49645454087898, 12.61588373354552, 13.13408564456123, 14.05215177993410

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