Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 79 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.67·2-s − 3-s + 5.14·4-s + 2.19·5-s − 2.67·6-s + 0.0367·7-s + 8.39·8-s + 9-s + 5.85·10-s − 2.40·11-s − 5.14·12-s + 3.84·13-s + 0.0980·14-s − 2.19·15-s + 12.1·16-s − 17-s + 2.67·18-s + 2.70·19-s + 11.2·20-s − 0.0367·21-s − 6.43·22-s + 2.52·23-s − 8.39·24-s − 0.200·25-s + 10.2·26-s − 27-s + 0.188·28-s + ⋯
L(s)  = 1  + 1.88·2-s − 0.577·3-s + 2.57·4-s + 0.979·5-s − 1.09·6-s + 0.0138·7-s + 2.96·8-s + 0.333·9-s + 1.85·10-s − 0.726·11-s − 1.48·12-s + 1.06·13-s + 0.0262·14-s − 0.565·15-s + 3.03·16-s − 0.242·17-s + 0.629·18-s + 0.619·19-s + 2.51·20-s − 0.00801·21-s − 1.37·22-s + 0.525·23-s − 1.71·24-s − 0.0400·25-s + 2.01·26-s − 0.192·27-s + 0.0356·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4029\)    =    \(3 \cdot 17 \cdot 79\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4029} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.984452033$
$L(\frac12)$  $\approx$  $6.984452033$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;17,\;79\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;79\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
79 \( 1 - T \)
good2 \( 1 - 2.67T + 2T^{2} \)
5 \( 1 - 2.19T + 5T^{2} \)
7 \( 1 - 0.0367T + 7T^{2} \)
11 \( 1 + 2.40T + 11T^{2} \)
13 \( 1 - 3.84T + 13T^{2} \)
19 \( 1 - 2.70T + 19T^{2} \)
23 \( 1 - 2.52T + 23T^{2} \)
29 \( 1 + 4.94T + 29T^{2} \)
31 \( 1 - 6.79T + 31T^{2} \)
37 \( 1 + 0.146T + 37T^{2} \)
41 \( 1 + 5.64T + 41T^{2} \)
43 \( 1 - 6.67T + 43T^{2} \)
47 \( 1 + 3.17T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 7.01T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 1.40T + 71T^{2} \)
73 \( 1 - 16.4T + 73T^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 - 8.20T + 97T^{2} \)
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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

−8.152097590167921854488251376716, −7.31423140717563532091554015301, −6.49600852259928339659118597348, −6.04378212203102840301866488874, −5.39117090945540810001958819158, −4.90043793938145835584929581296, −3.96939802430161256916436175138, −3.13290441334313013910690499468, −2.25689872180819293065909681341, −1.33195467785106109423253174321, 1.33195467785106109423253174321, 2.25689872180819293065909681341, 3.13290441334313013910690499468, 3.96939802430161256916436175138, 4.90043793938145835584929581296, 5.39117090945540810001958819158, 6.04378212203102840301866488874, 6.49600852259928339659118597348, 7.31423140717563532091554015301, 8.152097590167921854488251376716

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