Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 \cdot 31 $
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-s + 0.300·3-s + 4-s − 5-s + 0.300·6-s + 3.15·7-s + 8-s − 2.90·9-s − 10-s − 4.96·11-s + 0.300·12-s + 13-s + 3.15·14-s − 0.300·15-s + 16-s + 1.88·17-s − 2.90·18-s − 4.66·19-s − 20-s + 0.948·21-s − 4.96·22-s − 4.45·23-s + 0.300·24-s + 25-s + 26-s − 1.77·27-s + 3.15·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.173·3-s + 0.5·4-s − 0.447·5-s + 0.122·6-s + 1.19·7-s + 0.353·8-s − 0.969·9-s − 0.316·10-s − 1.49·11-s + 0.0868·12-s + 0.277·13-s + 0.841·14-s − 0.0776·15-s + 0.250·16-s + 0.458·17-s − 0.685·18-s − 1.07·19-s − 0.223·20-s + 0.206·21-s − 1.05·22-s − 0.929·23-s + 0.0614·24-s + 0.200·25-s + 0.196·26-s − 0.342·27-s + 0.595·28-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  =  1
Selberg data  =  $(2,\ 4030,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 - 0.300T + 3T^{2} \)
7 \( 1 - 3.15T + 7T^{2} \)
11 \( 1 + 4.96T + 11T^{2} \)
17 \( 1 - 1.88T + 17T^{2} \)
19 \( 1 + 4.66T + 19T^{2} \)
23 \( 1 + 4.45T + 23T^{2} \)
29 \( 1 + 2.08T + 29T^{2} \)
37 \( 1 + 5.70T + 37T^{2} \)
41 \( 1 - 6.80T + 41T^{2} \)
43 \( 1 + 0.777T + 43T^{2} \)
47 \( 1 + 6.80T + 47T^{2} \)
53 \( 1 + 3.39T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 + 3.23T + 61T^{2} \)
67 \( 1 + 8.44T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + 2.47T + 79T^{2} \)
83 \( 1 + 2.94T + 83T^{2} \)
89 \( 1 - 8.26T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−8.037142974616443782204719943764, −7.59516506564179656460095180798, −6.44947895264839429686267776816, −5.64434476296746216144719999851, −5.08417015658741956917256633796, −4.35015115593125417879829886458, −3.42838338611249459719349813222, −2.57120832499572523432133051955, −1.74589457789337261977549738655, 0, 1.74589457789337261977549738655, 2.57120832499572523432133051955, 3.42838338611249459719349813222, 4.35015115593125417879829886458, 5.08417015658741956917256633796, 5.64434476296746216144719999851, 6.44947895264839429686267776816, 7.59516506564179656460095180798, 8.037142974616443782204719943764

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