Properties

Label 2-56e2-1.1-c1-0-62
Degree $2$
Conductor $3136$
Sign $-1$
Analytic cond. $25.0410$
Root an. cond. $5.00410$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 3·9-s − 4·11-s + 2·13-s + 6·17-s − 8·19-s − 25-s − 6·29-s + 8·31-s + 2·37-s − 2·41-s − 4·43-s − 6·45-s − 8·47-s − 6·53-s − 8·55-s − 6·61-s + 4·65-s − 4·67-s + 8·71-s − 10·73-s − 16·79-s + 9·81-s − 8·83-s + 12·85-s + 6·89-s − 16·95-s + ⋯
L(s)  = 1  + 0.894·5-s − 9-s − 1.20·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.894·45-s − 1.16·47-s − 0.824·53-s − 1.07·55-s − 0.768·61-s + 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s − 1.80·79-s + 81-s − 0.878·83-s + 1.30·85-s + 0.635·89-s − 1.64·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(25.0410\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3136} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−8.227785865636410870393529644581, −7.84054312303988042972335609462, −6.59264336057149769206055975154, −5.91740132798549001211232285063, −5.47493819124631815597461836959, −4.53211800578457486655010876487, −3.32773603072291404011890923635, −2.58182243115007516792385368988, −1.62407385510972851799092712087, 0, 1.62407385510972851799092712087, 2.58182243115007516792385368988, 3.32773603072291404011890923635, 4.53211800578457486655010876487, 5.47493819124631815597461836959, 5.91740132798549001211232285063, 6.59264336057149769206055975154, 7.84054312303988042972335609462, 8.227785865636410870393529644581

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