Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 431 $
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 + 1.20·3-s + 4-s − 2.97·5-s − 1.20·6-s − 7-s − 8-s − 1.55·9-s + 2.97·10-s + 4.89·11-s + 1.20·12-s + 0.849·13-s + 14-s − 3.58·15-s + 16-s + 0.911·17-s + 1.55·18-s − 0.375·19-s − 2.97·20-s − 1.20·21-s − 4.89·22-s − 0.153·23-s − 1.20·24-s + 3.86·25-s − 0.849·26-s − 5.47·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.695·3-s + 0.5·4-s − 1.33·5-s − 0.491·6-s − 0.377·7-s − 0.353·8-s − 0.516·9-s + 0.941·10-s + 1.47·11-s + 0.347·12-s + 0.235·13-s + 0.267·14-s − 0.925·15-s + 0.250·16-s + 0.221·17-s + 0.365·18-s − 0.0860·19-s − 0.665·20-s − 0.262·21-s − 1.04·22-s − 0.0320·23-s − 0.245·24-s + 0.773·25-s − 0.166·26-s − 1.05·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6034\)    =    \(2 \cdot 7 \cdot 431\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.127709627$
$L(\frac12)$  $\approx$  $1.127709627$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;431\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;431\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
7 \( 1 + T \)
431 \( 1 - T \)
good3 \( 1 - 1.20T + 3T^{2} \)
5 \( 1 + 2.97T + 5T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - 0.849T + 13T^{2} \)
17 \( 1 - 0.911T + 17T^{2} \)
19 \( 1 + 0.375T + 19T^{2} \)
23 \( 1 + 0.153T + 23T^{2} \)
29 \( 1 - 6.12T + 29T^{2} \)
31 \( 1 - 5.66T + 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 + 3.62T + 41T^{2} \)
43 \( 1 + 0.855T + 43T^{2} \)
47 \( 1 + 8.00T + 47T^{2} \)
53 \( 1 + 3.06T + 53T^{2} \)
59 \( 1 - 9.27T + 59T^{2} \)
61 \( 1 - 6.50T + 61T^{2} \)
67 \( 1 + 4.31T + 67T^{2} \)
71 \( 1 + 0.739T + 71T^{2} \)
73 \( 1 + 8.19T + 73T^{2} \)
79 \( 1 - 17.0T + 79T^{2} \)
83 \( 1 - 6.32T + 83T^{2} \)
89 \( 1 - 0.101T + 89T^{2} \)
97 \( 1 - 4.01T + 97T^{2} \)
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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

−8.323082334966808012193369951410, −7.56736644782199162795820232767, −6.73495536718760827914883789491, −6.36726777821559744713262867858, −5.16220318600016970961408349363, −4.10410698534293828936972288985, −3.51490759081213547605565102781, −2.94316888933666644874609700143, −1.73762844614143891257788591078, −0.60072733068810213797099636593, 0.60072733068810213797099636593, 1.73762844614143891257788591078, 2.94316888933666644874609700143, 3.51490759081213547605565102781, 4.10410698534293828936972288985, 5.16220318600016970961408349363, 6.36726777821559744713262867858, 6.73495536718760827914883789491, 7.56736644782199162795820232767, 8.323082334966808012193369951410

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