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  + 2.29·2-s − 3-s + 3.25·4-s + 3.13·5-s − 2.29·6-s + 0.344·7-s + 2.86·8-s + 9-s + 7.18·10-s + 5.23·11-s − 3.25·12-s + 2.96·13-s + 0.789·14-s − 3.13·15-s + 0.0627·16-s − 17-s + 2.29·18-s − 2.69·19-s + 10.1·20-s − 0.344·21-s + 11.9·22-s + 7.52·23-s − 2.86·24-s + 4.83·25-s + 6.79·26-s − 27-s + 1.11·28-s + ⋯
L(s)  = 1  + 1.62·2-s − 0.577·3-s + 1.62·4-s + 1.40·5-s − 0.935·6-s + 0.130·7-s + 1.01·8-s + 0.333·9-s + 2.27·10-s + 1.57·11-s − 0.938·12-s + 0.822·13-s + 0.210·14-s − 0.809·15-s + 0.0156·16-s − 0.242·17-s + 0.540·18-s − 0.618·19-s + 2.27·20-s − 0.0751·21-s + 2.55·22-s + 1.56·23-s − 0.584·24-s + 0.967·25-s + 1.33·26-s − 0.192·27-s + 0.211·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.155026200$
$L(\frac12)$  $\approx$  $6.155026200$
$L(\frac{3}{2})$   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 \( 1 - 2.29T + 2T^{2} \)
5 \( 1 - 3.13T + 5T^{2} \)
7 \( 1 - 0.344T + 7T^{2} \)
11 \( 1 - 5.23T + 11T^{2} \)
13 \( 1 - 2.96T + 13T^{2} \)
19 \( 1 + 2.69T + 19T^{2} \)
23 \( 1 - 7.52T + 23T^{2} \)
29 \( 1 + 1.77T + 29T^{2} \)
31 \( 1 + 0.447T + 31T^{2} \)
37 \( 1 + 0.229T + 37T^{2} \)
41 \( 1 + 7.81T + 41T^{2} \)
43 \( 1 + 6.20T + 43T^{2} \)
47 \( 1 + 3.09T + 47T^{2} \)
53 \( 1 - 1.46T + 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 - 8.00T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 0.297T + 71T^{2} \)
73 \( 1 + 5.70T + 73T^{2} \)
83 \( 1 - 4.51T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−8.628377904274578813129201795738, −7.08734714880179482851263522986, −6.47477593055870477848570183705, −6.24746723934719662331075032142, −5.36840280932526333243320652525, −4.86029738508107795003595226851, −3.95681500953710041479356956020, −3.23036731785201653146022887994, −2.04580437997307290003746077004, −1.33247273421014261575785677066, 1.33247273421014261575785677066, 2.04580437997307290003746077004, 3.23036731785201653146022887994, 3.95681500953710041479356956020, 4.86029738508107795003595226851, 5.36840280932526333243320652525, 6.24746723934719662331075032142, 6.47477593055870477848570183705, 7.08734714880179482851263522986, 8.628377904274578813129201795738

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