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 − 0.0752·3-s + 4-s − 2.78·5-s + 0.0752·6-s − 7-s − 8-s − 2.99·9-s + 2.78·10-s + 2.08·11-s − 0.0752·12-s − 3.50·13-s + 14-s + 0.209·15-s + 16-s − 5.12·17-s + 2.99·18-s + 1.33·19-s − 2.78·20-s + 0.0752·21-s − 2.08·22-s − 0.969·23-s + 0.0752·24-s + 2.77·25-s + 3.50·26-s + 0.450·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0434·3-s + 0.5·4-s − 1.24·5-s + 0.0307·6-s − 0.377·7-s − 0.353·8-s − 0.998·9-s + 0.881·10-s + 0.630·11-s − 0.0217·12-s − 0.971·13-s + 0.267·14-s + 0.0541·15-s + 0.250·16-s − 1.24·17-s + 0.705·18-s + 0.305·19-s − 0.623·20-s + 0.0164·21-s − 0.445·22-s − 0.202·23-s + 0.0153·24-s + 0.555·25-s + 0.686·26-s + 0.0867·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1722820628$
$L(\frac12)$  $\approx$  $0.1722820628$
$L(\frac{3}{2})$   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 \( 1 + 0.0752T + 3T^{2} \)
5 \( 1 + 2.78T + 5T^{2} \)
11 \( 1 - 2.08T + 11T^{2} \)
13 \( 1 + 3.50T + 13T^{2} \)
17 \( 1 + 5.12T + 17T^{2} \)
19 \( 1 - 1.33T + 19T^{2} \)
23 \( 1 + 0.969T + 23T^{2} \)
29 \( 1 + 0.965T + 29T^{2} \)
31 \( 1 - 2.05T + 31T^{2} \)
37 \( 1 + 4.89T + 37T^{2} \)
41 \( 1 + 7.97T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 + 0.588T + 47T^{2} \)
53 \( 1 + 0.927T + 53T^{2} \)
59 \( 1 - 4.91T + 59T^{2} \)
61 \( 1 + 9.55T + 61T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 + 6.13T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 + 7.10T + 97T^{2} \)
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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

−8.197113425073264853068418773060, −7.40265712292987040324065043940, −6.84606068427823887238252193942, −6.19877781755302044246505861322, −5.17379105374688434046517474545, −4.39211894490977791433795562643, −3.47850262425585156520147933483, −2.84187740451512601593693608175, −1.75731492826400539770316054068, −0.23172483262533671477432803943, 0.23172483262533671477432803943, 1.75731492826400539770316054068, 2.84187740451512601593693608175, 3.47850262425585156520147933483, 4.39211894490977791433795562643, 5.17379105374688434046517474545, 6.19877781755302044246505861322, 6.84606068427823887238252193942, 7.40265712292987040324065043940, 8.197113425073264853068418773060

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