Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 \cdot 31 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3.08·3-s + 4-s − 5-s − 3.08·6-s − 2.62·7-s + 8-s + 6.50·9-s − 10-s + 0.190·11-s − 3.08·12-s − 13-s − 2.62·14-s + 3.08·15-s + 16-s − 6.93·17-s + 6.50·18-s − 0.237·19-s − 20-s + 8.09·21-s + 0.190·22-s + 4.00·23-s − 3.08·24-s + 25-s − 26-s − 10.7·27-s − 2.62·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.77·3-s + 0.5·4-s − 0.447·5-s − 1.25·6-s − 0.992·7-s + 0.353·8-s + 2.16·9-s − 0.316·10-s + 0.0573·11-s − 0.889·12-s − 0.277·13-s − 0.701·14-s + 0.795·15-s + 0.250·16-s − 1.68·17-s + 1.53·18-s − 0.0544·19-s − 0.223·20-s + 1.76·21-s + 0.0405·22-s + 0.835·23-s − 0.629·24-s + 0.200·25-s − 0.196·26-s − 2.07·27-s − 0.496·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4030} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4030,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7189092777$
$L(\frac12)$  $\approx$  $0.7189092777$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 + T \)
31 \( 1 + T \)
good3 \( 1 + 3.08T + 3T^{2} \)
7 \( 1 + 2.62T + 7T^{2} \)
11 \( 1 - 0.190T + 11T^{2} \)
17 \( 1 + 6.93T + 17T^{2} \)
19 \( 1 + 0.237T + 19T^{2} \)
23 \( 1 - 4.00T + 23T^{2} \)
29 \( 1 - 0.0461T + 29T^{2} \)
37 \( 1 + 9.02T + 37T^{2} \)
41 \( 1 + 2.50T + 41T^{2} \)
43 \( 1 + 2.72T + 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + 1.99T + 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 - 0.753T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 - 6.14T + 71T^{2} \)
73 \( 1 + 5.91T + 73T^{2} \)
79 \( 1 - 0.817T + 79T^{2} \)
83 \( 1 - 0.435T + 83T^{2} \)
89 \( 1 + 1.09T + 89T^{2} \)
97 \( 1 - 8.17T + 97T^{2} \)
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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

−8.357418008609312900637231586746, −7.10433163817356466791134675739, −6.76627781021298076786751457837, −6.32534813072699113273958828252, −5.33851195169697819947305796181, −4.86753671725306544566738345874, −4.07699460132144473485264224153, −3.22320382963540322605273198910, −1.90715147558623358983163568697, −0.46056767795535108364305203318, 0.46056767795535108364305203318, 1.90715147558623358983163568697, 3.22320382963540322605273198910, 4.07699460132144473485264224153, 4.86753671725306544566738345874, 5.33851195169697819947305796181, 6.32534813072699113273958828252, 6.76627781021298076786751457837, 7.10433163817356466791134675739, 8.357418008609312900637231586746

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