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  + 0.959·2-s + 3-s − 1.07·4-s − 3.93·5-s + 0.959·6-s − 0.0159·7-s − 2.95·8-s + 9-s − 3.77·10-s − 0.0651·11-s − 1.07·12-s − 1.13·13-s − 0.0153·14-s − 3.93·15-s − 0.675·16-s + 17-s + 0.959·18-s − 6.81·19-s + 4.24·20-s − 0.0159·21-s − 0.0625·22-s − 3.11·23-s − 2.95·24-s + 10.4·25-s − 1.09·26-s + 27-s + 0.0172·28-s + ⋯
L(s)  = 1  + 0.678·2-s + 0.577·3-s − 0.539·4-s − 1.75·5-s + 0.391·6-s − 0.00603·7-s − 1.04·8-s + 0.333·9-s − 1.19·10-s − 0.0196·11-s − 0.311·12-s − 0.316·13-s − 0.00409·14-s − 1.01·15-s − 0.168·16-s + 0.242·17-s + 0.226·18-s − 1.56·19-s + 0.948·20-s − 0.00348·21-s − 0.0133·22-s − 0.649·23-s − 0.603·24-s + 2.08·25-s − 0.214·26-s + 0.192·27-s + 0.00325·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.185875179$
$L(\frac12)$  $\approx$  $1.185875179$
$L(\frac{3}{2})$   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 \( 1 - 0.959T + 2T^{2} \)
5 \( 1 + 3.93T + 5T^{2} \)
7 \( 1 + 0.0159T + 7T^{2} \)
11 \( 1 + 0.0651T + 11T^{2} \)
13 \( 1 + 1.13T + 13T^{2} \)
19 \( 1 + 6.81T + 19T^{2} \)
23 \( 1 + 3.11T + 23T^{2} \)
29 \( 1 - 0.147T + 29T^{2} \)
31 \( 1 + 3.00T + 31T^{2} \)
37 \( 1 - 0.619T + 37T^{2} \)
41 \( 1 - 6.70T + 41T^{2} \)
43 \( 1 - 0.482T + 43T^{2} \)
47 \( 1 + 1.80T + 47T^{2} \)
53 \( 1 - 2.47T + 53T^{2} \)
59 \( 1 + 0.178T + 59T^{2} \)
61 \( 1 - 2.91T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 - 7.32T + 71T^{2} \)
73 \( 1 - 4.12T + 73T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 - 3.56T + 89T^{2} \)
97 \( 1 + 0.920T + 97T^{2} \)
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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

−8.199348081196381129920584672850, −8.000857272684094492500589236639, −7.03406146389583727188636369774, −6.25897201023587455290414605684, −5.15157410683414397267507957176, −4.41269174304413175921965026020, −3.90769239667842029242933795909, −3.33756141398344879899016041533, −2.30205534795012192206455928435, −0.52698195702404385626707144605, 0.52698195702404385626707144605, 2.30205534795012192206455928435, 3.33756141398344879899016041533, 3.90769239667842029242933795909, 4.41269174304413175921965026020, 5.15157410683414397267507957176, 6.25897201023587455290414605684, 7.03406146389583727188636369774, 8.000857272684094492500589236639, 8.199348081196381129920584672850

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