Properties

Degree 2
Conductor $ 29 \cdot 139 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.62·2-s + 0.935·3-s + 4.90·4-s − 2.36·5-s − 2.45·6-s + 0.343·7-s − 7.64·8-s − 2.12·9-s + 6.21·10-s + 4.42·11-s + 4.58·12-s − 2.43·13-s − 0.903·14-s − 2.21·15-s + 10.2·16-s + 5.05·17-s + 5.58·18-s − 1.41·19-s − 11.5·20-s + 0.321·21-s − 11.6·22-s + 1.96·23-s − 7.14·24-s + 0.585·25-s + 6.40·26-s − 4.79·27-s + 1.68·28-s + ⋯
L(s)  = 1  − 1.85·2-s + 0.539·3-s + 2.45·4-s − 1.05·5-s − 1.00·6-s + 0.129·7-s − 2.70·8-s − 0.708·9-s + 1.96·10-s + 1.33·11-s + 1.32·12-s − 0.675·13-s − 0.241·14-s − 0.570·15-s + 2.56·16-s + 1.22·17-s + 1.31·18-s − 0.324·19-s − 2.59·20-s + 0.0701·21-s − 2.48·22-s + 0.409·23-s − 1.45·24-s + 0.117·25-s + 1.25·26-s − 0.922·27-s + 0.319·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4031\)    =    \(29 \cdot 139\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4031} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4031,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{29,\;139\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{29,\;139\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad29 \( 1 + T \)
139 \( 1 + T \)
good2 \( 1 + 2.62T + 2T^{2} \)
3 \( 1 - 0.935T + 3T^{2} \)
5 \( 1 + 2.36T + 5T^{2} \)
7 \( 1 - 0.343T + 7T^{2} \)
11 \( 1 - 4.42T + 11T^{2} \)
13 \( 1 + 2.43T + 13T^{2} \)
17 \( 1 - 5.05T + 17T^{2} \)
19 \( 1 + 1.41T + 19T^{2} \)
23 \( 1 - 1.96T + 23T^{2} \)
31 \( 1 - 0.410T + 31T^{2} \)
37 \( 1 + 1.20T + 37T^{2} \)
41 \( 1 + 8.28T + 41T^{2} \)
43 \( 1 + 1.27T + 43T^{2} \)
47 \( 1 + 1.89T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 - 8.43T + 59T^{2} \)
61 \( 1 + 4.77T + 61T^{2} \)
67 \( 1 + 3.01T + 67T^{2} \)
71 \( 1 - 8.62T + 71T^{2} \)
73 \( 1 - 1.28T + 73T^{2} \)
79 \( 1 - 1.49T + 79T^{2} \)
83 \( 1 - 3.80T + 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + 2.47T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.207618456443228046126900941264, −7.66524667726498539968634364958, −7.01076235832735585525500253925, −6.31870303825831360810343582563, −5.22931449762692055211449108210, −3.83842533399637347653943289638, −3.20646543614116045786723034985, −2.19026901232322368454984135696, −1.15321172579857339528896298780, 0, 1.15321172579857339528896298780, 2.19026901232322368454984135696, 3.20646543614116045786723034985, 3.83842533399637347653943289638, 5.22931449762692055211449108210, 6.31870303825831360810343582563, 7.01076235832735585525500253925, 7.66524667726498539968634364958, 8.207618456443228046126900941264

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