Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 13 \cdot 103 $
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-s + 4-s + 3.53·5-s − 6-s − 4.09·7-s + 8-s + 9-s + 3.53·10-s − 3.53·11-s − 12-s − 13-s − 4.09·14-s − 3.53·15-s + 16-s − 2.18·17-s + 18-s − 0.266·19-s + 3.53·20-s + 4.09·21-s − 3.53·22-s − 9.08·23-s − 24-s + 7.51·25-s − 26-s − 27-s − 4.09·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.58·5-s − 0.408·6-s − 1.54·7-s + 0.353·8-s + 0.333·9-s + 1.11·10-s − 1.06·11-s − 0.288·12-s − 0.277·13-s − 1.09·14-s − 0.913·15-s + 0.250·16-s − 0.530·17-s + 0.235·18-s − 0.0611·19-s + 0.791·20-s + 0.894·21-s − 0.752·22-s − 1.89·23-s − 0.204·24-s + 1.50·25-s − 0.196·26-s − 0.192·27-s − 0.774·28-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  =  \(0\)
Selberg data  =  \((2,\ 8034,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.481538346\)
\(L(\frac12)\)  \(\approx\)  \(2.481538346\)
\(L(\frac{3}{2})\)   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 \( 1 - 3.53T + 5T^{2} \)
7 \( 1 + 4.09T + 7T^{2} \)
11 \( 1 + 3.53T + 11T^{2} \)
17 \( 1 + 2.18T + 17T^{2} \)
19 \( 1 + 0.266T + 19T^{2} \)
23 \( 1 + 9.08T + 23T^{2} \)
29 \( 1 - 6.77T + 29T^{2} \)
31 \( 1 - 6.69T + 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + 5.22T + 41T^{2} \)
43 \( 1 - 6.71T + 43T^{2} \)
47 \( 1 + 8.08T + 47T^{2} \)
53 \( 1 - 5.93T + 53T^{2} \)
59 \( 1 - 1.22T + 59T^{2} \)
61 \( 1 + 4.83T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 8.30T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 - 8.26T + 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 8.23T + 97T^{2} \)
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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

−7.65078063197255575352670265789, −6.63511814559977180619135249694, −6.26424127695961172520200469731, −5.96099384019822644089857323908, −5.12397201787048931694869799370, −4.50085345420321054892072732120, −3.45135380163895138748984970414, −2.52524445847354455206035970564, −2.17833954194469145942650872068, −0.68381336246189624977667748447, 0.68381336246189624977667748447, 2.17833954194469145942650872068, 2.52524445847354455206035970564, 3.45135380163895138748984970414, 4.50085345420321054892072732120, 5.12397201787048931694869799370, 5.96099384019822644089857323908, 6.26424127695961172520200469731, 6.63511814559977180619135249694, 7.65078063197255575352670265789

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