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 − 1.92·3-s + 4-s − 2.53·5-s + 1.92·6-s − 7-s − 8-s + 0.687·9-s + 2.53·10-s − 3.57·11-s − 1.92·12-s + 3.61·13-s + 14-s + 4.86·15-s + 16-s − 0.257·17-s − 0.687·18-s + 0.598·19-s − 2.53·20-s + 1.92·21-s + 3.57·22-s + 8.37·23-s + 1.92·24-s + 1.40·25-s − 3.61·26-s + 4.44·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.10·3-s + 0.5·4-s − 1.13·5-s + 0.783·6-s − 0.377·7-s − 0.353·8-s + 0.229·9-s + 0.800·10-s − 1.07·11-s − 0.554·12-s + 1.00·13-s + 0.267·14-s + 1.25·15-s + 0.250·16-s − 0.0623·17-s − 0.162·18-s + 0.137·19-s − 0.566·20-s + 0.419·21-s + 0.763·22-s + 1.74·23-s + 0.391·24-s + 0.281·25-s − 0.709·26-s + 0.854·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2153487476$
$L(\frac12)$  $\approx$  $0.2153487476$
$L(\frac{3}{2})$   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 \( 1 + 1.92T + 3T^{2} \)
5 \( 1 + 2.53T + 5T^{2} \)
11 \( 1 + 3.57T + 11T^{2} \)
13 \( 1 - 3.61T + 13T^{2} \)
17 \( 1 + 0.257T + 17T^{2} \)
19 \( 1 - 0.598T + 19T^{2} \)
23 \( 1 - 8.37T + 23T^{2} \)
29 \( 1 + 4.10T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 4.11T + 37T^{2} \)
41 \( 1 - 2.61T + 41T^{2} \)
43 \( 1 + 2.17T + 43T^{2} \)
47 \( 1 - 4.93T + 47T^{2} \)
53 \( 1 - 1.94T + 53T^{2} \)
59 \( 1 + 8.21T + 59T^{2} \)
61 \( 1 + 15.5T + 61T^{2} \)
67 \( 1 + 8.51T + 67T^{2} \)
71 \( 1 + 4.42T + 71T^{2} \)
73 \( 1 + 9.04T + 73T^{2} \)
79 \( 1 + 2.43T + 79T^{2} \)
83 \( 1 - 1.85T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−7.930617549856609236560227131437, −7.40690892711693862447546400333, −6.82184326385492698090029433996, −5.89635575900893733027845106221, −5.42781364713097667433602252537, −4.53960915920288245826475347995, −3.52900098382424429722885788967, −2.88573399828676190843862034196, −1.44058778958727662530046955988, −0.29895669106892863316162302198, 0.29895669106892863316162302198, 1.44058778958727662530046955988, 2.88573399828676190843862034196, 3.52900098382424429722885788967, 4.53960915920288245826475347995, 5.42781364713097667433602252537, 5.89635575900893733027845106221, 6.82184326385492698090029433996, 7.40690892711693862447546400333, 7.930617549856609236560227131437

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