Properties

Label 2-12696-1.1-c1-0-3
Degree $2$
Conductor $12696$
Sign $1$
Analytic cond. $101.378$
Root an. cond. $10.0686$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s + 9-s + 2·13-s − 8·17-s − 6·19-s − 2·21-s − 5·25-s − 27-s + 2·29-s − 4·31-s − 6·37-s − 2·39-s + 10·41-s − 6·43-s − 3·49-s + 8·51-s − 12·53-s + 6·57-s + 4·59-s + 10·61-s + 2·63-s + 6·67-s + 2·73-s + 5·75-s + 6·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.554·13-s − 1.94·17-s − 1.37·19-s − 0.436·21-s − 25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.986·37-s − 0.320·39-s + 1.56·41-s − 0.914·43-s − 3/7·49-s + 1.12·51-s − 1.64·53-s + 0.794·57-s + 0.520·59-s + 1.28·61-s + 0.251·63-s + 0.733·67-s + 0.234·73-s + 0.577·75-s + 0.675·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12696\)    =    \(2^{3} \cdot 3 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(101.378\)
Root analytic conductor: \(10.0686\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12696} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 12696,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.219267819\)
\(L(\frac12)\) \(\approx\) \(1.219267819\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
23 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−16.17954043027034, −15.75660551522661, −15.28685554667838, −14.61936643093179, −14.06951157136707, −13.34533967204618, −12.91828320655161, −12.36300874282211, −11.45549914193290, −11.20057123049736, −10.76855893648890, −10.07058751432795, −9.269539202888082, −8.648953840011464, −8.195358075327149, −7.417687415691449, −6.590934738267347, −6.299777130657663, −5.455185576985623, −4.690836356716176, −4.265428539926763, −3.479100586022439, −2.153498094109886, −1.838700151643645, −0.4981462155510297, 0.4981462155510297, 1.838700151643645, 2.153498094109886, 3.479100586022439, 4.265428539926763, 4.690836356716176, 5.455185576985623, 6.299777130657663, 6.590934738267347, 7.417687415691449, 8.195358075327149, 8.648953840011464, 9.269539202888082, 10.07058751432795, 10.76855893648890, 11.20057123049736, 11.45549914193290, 12.36300874282211, 12.91828320655161, 13.34533967204618, 14.06951157136707, 14.61936643093179, 15.28685554667838, 15.75660551522661, 16.17954043027034

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