Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.86·3-s − 0.0527·5-s − 1.89·7-s + 5.19·9-s − 5.46·11-s − 0.847·13-s − 0.151·15-s − 17-s + 0.788·19-s − 5.42·21-s + 0.154·23-s − 4.99·25-s + 6.29·27-s − 3.17·29-s − 8.43·31-s − 15.6·33-s + 0.100·35-s − 3.87·37-s − 2.42·39-s − 1.73·41-s − 2.07·43-s − 0.274·45-s + 3.26·47-s − 3.40·49-s − 2.86·51-s + 2.52·53-s + 0.288·55-s + ⋯
L(s)  = 1  + 1.65·3-s − 0.0236·5-s − 0.716·7-s + 1.73·9-s − 1.64·11-s − 0.235·13-s − 0.0390·15-s − 0.242·17-s + 0.181·19-s − 1.18·21-s + 0.0322·23-s − 0.999·25-s + 1.21·27-s − 0.588·29-s − 1.51·31-s − 2.72·33-s + 0.0169·35-s − 0.637·37-s − 0.388·39-s − 0.270·41-s − 0.316·43-s − 0.0409·45-s + 0.475·47-s − 0.487·49-s − 0.400·51-s + 0.346·53-s + 0.0389·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 - T \)
good3 \( 1 - 2.86T + 3T^{2} \)
5 \( 1 + 0.0527T + 5T^{2} \)
7 \( 1 + 1.89T + 7T^{2} \)
11 \( 1 + 5.46T + 11T^{2} \)
13 \( 1 + 0.847T + 13T^{2} \)
19 \( 1 - 0.788T + 19T^{2} \)
23 \( 1 - 0.154T + 23T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 + 8.43T + 31T^{2} \)
37 \( 1 + 3.87T + 37T^{2} \)
41 \( 1 + 1.73T + 41T^{2} \)
43 \( 1 + 2.07T + 43T^{2} \)
47 \( 1 - 3.26T + 47T^{2} \)
53 \( 1 - 2.52T + 53T^{2} \)
61 \( 1 - 8.67T + 61T^{2} \)
67 \( 1 + 6.26T + 67T^{2} \)
71 \( 1 - 1.81T + 71T^{2} \)
73 \( 1 - 4.69T + 73T^{2} \)
79 \( 1 - 3.52T + 79T^{2} \)
83 \( 1 + 6.16T + 83T^{2} \)
89 \( 1 + 8.42T + 89T^{2} \)
97 \( 1 - 6.52T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.082067733053969298351763296607, −7.51410075862935183523890680076, −6.97171510187242170993076114729, −5.79481497243760817669362594423, −5.03490317891756573592136895756, −3.91120787083792045548731595635, −3.32511985107230159937620994411, −2.53745503599766375077134988684, −1.88214908499253841231632912509, 0, 1.88214908499253841231632912509, 2.53745503599766375077134988684, 3.32511985107230159937620994411, 3.91120787083792045548731595635, 5.03490317891756573592136895756, 5.79481497243760817669362594423, 6.97171510187242170993076114729, 7.51410075862935183523890680076, 8.082067733053969298351763296607

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