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  − 0.0772·2-s − 3-s − 1.99·4-s + 3.97·5-s + 0.0772·6-s + 0.969·7-s + 0.308·8-s + 9-s − 0.306·10-s + 2.24·11-s + 1.99·12-s + 4.20·13-s − 0.0748·14-s − 3.97·15-s + 3.96·16-s − 17-s − 0.0772·18-s + 5.15·19-s − 7.92·20-s − 0.969·21-s − 0.173·22-s + 0.911·23-s − 0.308·24-s + 10.7·25-s − 0.324·26-s − 27-s − 1.93·28-s + ⋯
L(s)  = 1  − 0.0546·2-s − 0.577·3-s − 0.997·4-s + 1.77·5-s + 0.0315·6-s + 0.366·7-s + 0.109·8-s + 0.333·9-s − 0.0970·10-s + 0.676·11-s + 0.575·12-s + 1.16·13-s − 0.0200·14-s − 1.02·15-s + 0.991·16-s − 0.242·17-s − 0.0182·18-s + 1.18·19-s − 1.77·20-s − 0.211·21-s − 0.0369·22-s + 0.189·23-s − 0.0629·24-s + 2.15·25-s − 0.0636·26-s − 0.192·27-s − 0.365·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.213154353$
$L(\frac12)$  $\approx$  $2.213154353$
$L(\frac{3}{2})$   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 \( 1 + 0.0772T + 2T^{2} \)
5 \( 1 - 3.97T + 5T^{2} \)
7 \( 1 - 0.969T + 7T^{2} \)
11 \( 1 - 2.24T + 11T^{2} \)
13 \( 1 - 4.20T + 13T^{2} \)
19 \( 1 - 5.15T + 19T^{2} \)
23 \( 1 - 0.911T + 23T^{2} \)
29 \( 1 - 8.81T + 29T^{2} \)
31 \( 1 + 5.98T + 31T^{2} \)
37 \( 1 - 4.64T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 9.32T + 43T^{2} \)
47 \( 1 - 6.28T + 47T^{2} \)
53 \( 1 + 4.67T + 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 + 7.64T + 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 - 0.523T + 71T^{2} \)
73 \( 1 + 9.82T + 73T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−8.691515450714350350629889193752, −7.80445035804982611238494032939, −6.66512571819539547699275226923, −6.14719516795873069253144472674, −5.43681448137698381946919500933, −4.92609121977346670628055455354, −4.00568193399757407858600371725, −2.95126831512056726536120696296, −1.57729895695523216901888319583, −1.03336612488968903161529614059, 1.03336612488968903161529614059, 1.57729895695523216901888319583, 2.95126831512056726536120696296, 4.00568193399757407858600371725, 4.92609121977346670628055455354, 5.43681448137698381946919500933, 6.14719516795873069253144472674, 6.66512571819539547699275226923, 7.80445035804982611238494032939, 8.691515450714350350629889193752

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