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 + 0.0287·3-s + 4-s + 1.06·5-s − 0.0287·6-s − 7-s − 8-s − 2.99·9-s − 1.06·10-s + 2.45·11-s + 0.0287·12-s − 6.62·13-s + 14-s + 0.0305·15-s + 16-s + 4.12·17-s + 2.99·18-s − 6.10·19-s + 1.06·20-s − 0.0287·21-s − 2.45·22-s − 5.32·23-s − 0.0287·24-s − 3.87·25-s + 6.62·26-s − 0.172·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.0165·3-s + 0.5·4-s + 0.475·5-s − 0.0117·6-s − 0.377·7-s − 0.353·8-s − 0.999·9-s − 0.335·10-s + 0.741·11-s + 0.00829·12-s − 1.83·13-s + 0.267·14-s + 0.00787·15-s + 0.250·16-s + 1.00·17-s + 0.706·18-s − 1.40·19-s + 0.237·20-s − 0.00626·21-s − 0.524·22-s − 1.11·23-s − 0.00586·24-s − 0.774·25-s + 1.29·26-s − 0.0331·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8468307598$
$L(\frac12)$  $\approx$  $0.8468307598$
$L(\frac{3}{2})$   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 \( 1 - 0.0287T + 3T^{2} \)
5 \( 1 - 1.06T + 5T^{2} \)
11 \( 1 - 2.45T + 11T^{2} \)
13 \( 1 + 6.62T + 13T^{2} \)
17 \( 1 - 4.12T + 17T^{2} \)
19 \( 1 + 6.10T + 19T^{2} \)
23 \( 1 + 5.32T + 23T^{2} \)
29 \( 1 + 4.61T + 29T^{2} \)
31 \( 1 - 9.64T + 31T^{2} \)
37 \( 1 - 5.32T + 37T^{2} \)
41 \( 1 + 8.80T + 41T^{2} \)
43 \( 1 - 1.40T + 43T^{2} \)
47 \( 1 + 1.51T + 47T^{2} \)
53 \( 1 - 6.97T + 53T^{2} \)
59 \( 1 - 1.91T + 59T^{2} \)
61 \( 1 - 8.07T + 61T^{2} \)
67 \( 1 - 4.82T + 67T^{2} \)
71 \( 1 + 3.67T + 71T^{2} \)
73 \( 1 - 2.42T + 73T^{2} \)
79 \( 1 - 1.75T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 5.39T + 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−8.145741922826750712103262372052, −7.49306738301382425293077249839, −6.63508493350725065772252193910, −6.07529378732125617602890863375, −5.43244627302097964339300154283, −4.43925962443540594902459621831, −3.47459871674822685228299103197, −2.49086899465120371063305842673, −1.98346733470570395259690044615, −0.50653418709760305676460211404, 0.50653418709760305676460211404, 1.98346733470570395259690044615, 2.49086899465120371063305842673, 3.47459871674822685228299103197, 4.43925962443540594902459621831, 5.43244627302097964339300154283, 6.07529378732125617602890863375, 6.63508493350725065772252193910, 7.49306738301382425293077249839, 8.145741922826750712103262372052

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