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 − 2.58·3-s + 4-s − 5-s − 2.58·6-s − 0.667·7-s + 8-s + 3.67·9-s − 10-s − 2.84·11-s − 2.58·12-s + 13-s − 0.667·14-s + 2.58·15-s + 16-s + 1.37·17-s + 3.67·18-s + 1.12·19-s − 20-s + 1.72·21-s − 2.84·22-s − 0.435·23-s − 2.58·24-s + 25-s + 26-s − 1.74·27-s − 0.667·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.49·3-s + 0.5·4-s − 0.447·5-s − 1.05·6-s − 0.252·7-s + 0.353·8-s + 1.22·9-s − 0.316·10-s − 0.858·11-s − 0.745·12-s + 0.277·13-s − 0.178·14-s + 0.667·15-s + 0.250·16-s + 0.333·17-s + 0.866·18-s + 0.258·19-s − 0.223·20-s + 0.376·21-s − 0.606·22-s − 0.0908·23-s − 0.527·24-s + 0.200·25-s + 0.196·26-s − 0.335·27-s − 0.126·28-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 \( 1 + 2.58T + 3T^{2} \)
7 \( 1 + 0.667T + 7T^{2} \)
11 \( 1 + 2.84T + 11T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 + 0.435T + 23T^{2} \)
29 \( 1 - 3.39T + 29T^{2} \)
37 \( 1 - 6.66T + 37T^{2} \)
41 \( 1 + 4.55T + 41T^{2} \)
43 \( 1 + 8.44T + 43T^{2} \)
47 \( 1 - 1.87T + 47T^{2} \)
53 \( 1 - 6.09T + 53T^{2} \)
59 \( 1 - 9.16T + 59T^{2} \)
61 \( 1 - 0.621T + 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 + 0.573T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + 8.71T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 - 0.261T + 89T^{2} \)
97 \( 1 + 14.0T + 97T^{2} \)
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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

−7.85613470943149432610925366501, −7.07568853474646279051724983248, −6.43908347625555048483282982968, −5.73820840724487134521860758098, −5.14607117783374098453494832861, −4.51473091711273147743596962466, −3.58381008013633195530218147078, −2.64860878177217478320781577730, −1.22298998758999407745186170780, 0, 1.22298998758999407745186170780, 2.64860878177217478320781577730, 3.58381008013633195530218147078, 4.51473091711273147743596962466, 5.14607117783374098453494832861, 5.73820840724487134521860758098, 6.43908347625555048483282982968, 7.07568853474646279051724983248, 7.85613470943149432610925366501

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