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 + 1.44·3-s + 4-s − 1.26·5-s − 1.44·6-s − 7-s − 8-s − 0.901·9-s + 1.26·10-s − 3.22·11-s + 1.44·12-s − 2.31·13-s + 14-s − 1.83·15-s + 16-s + 3.60·17-s + 0.901·18-s + 5.33·19-s − 1.26·20-s − 1.44·21-s + 3.22·22-s − 7.66·23-s − 1.44·24-s − 3.40·25-s + 2.31·26-s − 5.65·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.836·3-s + 0.5·4-s − 0.565·5-s − 0.591·6-s − 0.377·7-s − 0.353·8-s − 0.300·9-s + 0.399·10-s − 0.972·11-s + 0.418·12-s − 0.642·13-s + 0.267·14-s − 0.472·15-s + 0.250·16-s + 0.873·17-s + 0.212·18-s + 1.22·19-s − 0.282·20-s − 0.316·21-s + 0.687·22-s − 1.59·23-s − 0.295·24-s − 0.680·25-s + 0.454·26-s − 1.08·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9707624933$
$L(\frac12)$  $\approx$  $0.9707624933$
$L(\frac{3}{2})$   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 \( 1 - 1.44T + 3T^{2} \)
5 \( 1 + 1.26T + 5T^{2} \)
11 \( 1 + 3.22T + 11T^{2} \)
13 \( 1 + 2.31T + 13T^{2} \)
17 \( 1 - 3.60T + 17T^{2} \)
19 \( 1 - 5.33T + 19T^{2} \)
23 \( 1 + 7.66T + 23T^{2} \)
29 \( 1 + 0.214T + 29T^{2} \)
31 \( 1 + 5.47T + 31T^{2} \)
37 \( 1 - 0.154T + 37T^{2} \)
41 \( 1 - 3.69T + 41T^{2} \)
43 \( 1 - 7.50T + 43T^{2} \)
47 \( 1 - 0.101T + 47T^{2} \)
53 \( 1 - 9.39T + 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 - 5.95T + 61T^{2} \)
67 \( 1 + 2.95T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 15.7T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + 1.96T + 83T^{2} \)
89 \( 1 + 0.350T + 89T^{2} \)
97 \( 1 + 17.6T + 97T^{2} \)
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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

−8.016825818473592256024889582892, −7.56421415438349822257873881196, −7.15590239960064867187984817966, −5.71819653511185232793397119883, −5.58219117341381144345315401901, −4.15843072055506334389954040160, −3.43835579637760336759956630560, −2.71368245027600069458846135677, −2.00533838834854344082608309406, −0.52105020623183248076912944279, 0.52105020623183248076912944279, 2.00533838834854344082608309406, 2.71368245027600069458846135677, 3.43835579637760336759956630560, 4.15843072055506334389954040160, 5.58219117341381144345315401901, 5.71819653511185232793397119883, 7.15590239960064867187984817966, 7.56421415438349822257873881196, 8.016825818473592256024889582892

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