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  − 1.27·2-s + 3-s − 0.375·4-s − 3.50·5-s − 1.27·6-s − 0.192·7-s + 3.02·8-s + 9-s + 4.46·10-s + 3.68·11-s − 0.375·12-s − 3.36·13-s + 0.244·14-s − 3.50·15-s − 3.10·16-s + 17-s − 1.27·18-s + 6.71·19-s + 1.31·20-s − 0.192·21-s − 4.69·22-s + 6.93·23-s + 3.02·24-s + 7.27·25-s + 4.29·26-s + 27-s + 0.0722·28-s + ⋯
L(s)  = 1  − 0.901·2-s + 0.577·3-s − 0.187·4-s − 1.56·5-s − 0.520·6-s − 0.0726·7-s + 1.07·8-s + 0.333·9-s + 1.41·10-s + 1.11·11-s − 0.108·12-s − 0.934·13-s + 0.0654·14-s − 0.904·15-s − 0.776·16-s + 0.242·17-s − 0.300·18-s + 1.54·19-s + 0.294·20-s − 0.0419·21-s − 1.00·22-s + 1.44·23-s + 0.618·24-s + 1.45·25-s + 0.842·26-s + 0.192·27-s + 0.0136·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8590423123$
$L(\frac12)$  $\approx$  $0.8590423123$
$L(\frac{3}{2})$   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 \( 1 + 1.27T + 2T^{2} \)
5 \( 1 + 3.50T + 5T^{2} \)
7 \( 1 + 0.192T + 7T^{2} \)
11 \( 1 - 3.68T + 11T^{2} \)
13 \( 1 + 3.36T + 13T^{2} \)
19 \( 1 - 6.71T + 19T^{2} \)
23 \( 1 - 6.93T + 23T^{2} \)
29 \( 1 + 7.85T + 29T^{2} \)
31 \( 1 - 6.58T + 31T^{2} \)
37 \( 1 + 4.80T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 - 6.12T + 47T^{2} \)
53 \( 1 - 5.67T + 53T^{2} \)
59 \( 1 + 4.78T + 59T^{2} \)
61 \( 1 + 8.22T + 61T^{2} \)
67 \( 1 - 9.89T + 67T^{2} \)
71 \( 1 + 9.09T + 71T^{2} \)
73 \( 1 - 14.6T + 73T^{2} \)
83 \( 1 + 4.79T + 83T^{2} \)
89 \( 1 - 18.0T + 89T^{2} \)
97 \( 1 + 9.13T + 97T^{2} \)
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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

−8.439448598063316678203247248934, −7.83078523795946729767692162355, −7.22398963577829980977119020469, −6.82511090002457473215495827486, −5.15121940814726452778290027807, −4.63154018759628326963537301488, −3.62805437340513802259832660707, −3.21495321445785636169755060195, −1.63560205013491999934361465015, −0.62367107843450142498051304914, 0.62367107843450142498051304914, 1.63560205013491999934361465015, 3.21495321445785636169755060195, 3.62805437340513802259832660707, 4.63154018759628326963537301488, 5.15121940814726452778290027807, 6.82511090002457473215495827486, 7.22398963577829980977119020469, 7.83078523795946729767692162355, 8.439448598063316678203247248934

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