Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 13 \cdot 103 $
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 + 3-s + 4-s + 2.09·5-s − 6-s − 0.0373·7-s − 8-s + 9-s − 2.09·10-s − 3.17·11-s + 12-s + 13-s + 0.0373·14-s + 2.09·15-s + 16-s − 2.38·17-s − 18-s + 2.91·19-s + 2.09·20-s − 0.0373·21-s + 3.17·22-s + 3.99·23-s − 24-s − 0.601·25-s − 26-s + 27-s − 0.0373·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.937·5-s − 0.408·6-s − 0.0140·7-s − 0.353·8-s + 0.333·9-s − 0.663·10-s − 0.958·11-s + 0.288·12-s + 0.277·13-s + 0.00996·14-s + 0.541·15-s + 0.250·16-s − 0.577·17-s − 0.235·18-s + 0.668·19-s + 0.468·20-s − 0.00813·21-s + 0.677·22-s + 0.832·23-s − 0.204·24-s − 0.120·25-s − 0.196·26-s + 0.192·27-s − 0.00704·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 8034,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;13,\;103\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;13,\;103\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
13 \( 1 - T \)
103 \( 1 + T \)
good5 \( 1 - 2.09T + 5T^{2} \)
7 \( 1 + 0.0373T + 7T^{2} \)
11 \( 1 + 3.17T + 11T^{2} \)
17 \( 1 + 2.38T + 17T^{2} \)
19 \( 1 - 2.91T + 19T^{2} \)
23 \( 1 - 3.99T + 23T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 - 0.760T + 31T^{2} \)
37 \( 1 + 8.84T + 37T^{2} \)
41 \( 1 + 9.70T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + 5.29T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 4.58T + 59T^{2} \)
61 \( 1 + 1.68T + 61T^{2} \)
67 \( 1 - 7.20T + 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 - 6.39T + 79T^{2} \)
83 \( 1 + 5.88T + 83T^{2} \)
89 \( 1 - 2.63T + 89T^{2} \)
97 \( 1 + 2.73T + 97T^{2} \)
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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

−7.65610386052699384228678625914, −6.79669323311729809502431686031, −6.37844007471261079093207746352, −5.27341410953178123807045331365, −4.97363755756082296965652618827, −3.54836820420986416546378792305, −2.98897200129887899039740223719, −2.03298156282634615004203543434, −1.49447603746591919225882586726, 0, 1.49447603746591919225882586726, 2.03298156282634615004203543434, 2.98897200129887899039740223719, 3.54836820420986416546378792305, 4.97363755756082296965652618827, 5.27341410953178123807045331365, 6.37844007471261079093207746352, 6.79669323311729809502431686031, 7.65610386052699384228678625914

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