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 + 2.43·3-s + 4-s + 0.943·5-s − 2.43·6-s − 7-s − 8-s + 2.95·9-s − 0.943·10-s + 0.679·11-s + 2.43·12-s + 6.21·13-s + 14-s + 2.30·15-s + 16-s + 5.66·17-s − 2.95·18-s + 5.13·19-s + 0.943·20-s − 2.43·21-s − 0.679·22-s − 1.80·23-s − 2.43·24-s − 4.10·25-s − 6.21·26-s − 0.113·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.40·3-s + 0.5·4-s + 0.422·5-s − 0.996·6-s − 0.377·7-s − 0.353·8-s + 0.984·9-s − 0.298·10-s + 0.204·11-s + 0.704·12-s + 1.72·13-s + 0.267·14-s + 0.594·15-s + 0.250·16-s + 1.37·17-s − 0.696·18-s + 1.17·19-s + 0.211·20-s − 0.532·21-s − 0.144·22-s − 0.375·23-s − 0.498·24-s − 0.821·25-s − 1.21·26-s − 0.0218·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.088371991$
$L(\frac12)$  $\approx$  $3.088371991$
$L(\frac{3}{2})$   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 \( 1 - 2.43T + 3T^{2} \)
5 \( 1 - 0.943T + 5T^{2} \)
11 \( 1 - 0.679T + 11T^{2} \)
13 \( 1 - 6.21T + 13T^{2} \)
17 \( 1 - 5.66T + 17T^{2} \)
19 \( 1 - 5.13T + 19T^{2} \)
23 \( 1 + 1.80T + 23T^{2} \)
29 \( 1 - 2.77T + 29T^{2} \)
31 \( 1 - 2.08T + 31T^{2} \)
37 \( 1 + 5.86T + 37T^{2} \)
41 \( 1 - 3.44T + 41T^{2} \)
43 \( 1 + 3.18T + 43T^{2} \)
47 \( 1 - 6.61T + 47T^{2} \)
53 \( 1 + 1.94T + 53T^{2} \)
59 \( 1 + 6.17T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 2.57T + 67T^{2} \)
71 \( 1 + 2.07T + 71T^{2} \)
73 \( 1 - 0.542T + 73T^{2} \)
79 \( 1 + 1.17T + 79T^{2} \)
83 \( 1 - 4.51T + 83T^{2} \)
89 \( 1 + 18.7T + 89T^{2} \)
97 \( 1 + 5.15T + 97T^{2} \)
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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

−8.201148276765890387594309033950, −7.65514005230121748279133633249, −6.88255359298028616338139027189, −6.00730329408990104804905670688, −5.47984805538961162564025534349, −3.99431359365017487032270563266, −3.42094317307620548814009231665, −2.82553062312371782620075040199, −1.76705178623558999758949532356, −1.04343208229475818274465379360, 1.04343208229475818274465379360, 1.76705178623558999758949532356, 2.82553062312371782620075040199, 3.42094317307620548814009231665, 3.99431359365017487032270563266, 5.47984805538961162564025534349, 6.00730329408990104804905670688, 6.88255359298028616338139027189, 7.65514005230121748279133633249, 8.201148276765890387594309033950

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