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  + 1.38·2-s − 3-s − 0.0785·4-s + 0.223·5-s − 1.38·6-s − 1.36·7-s − 2.88·8-s + 9-s + 0.310·10-s − 1.81·11-s + 0.0785·12-s − 6.19·13-s − 1.88·14-s − 0.223·15-s − 3.83·16-s − 17-s + 1.38·18-s + 0.302·19-s − 0.0175·20-s + 1.36·21-s − 2.51·22-s + 3.55·23-s + 2.88·24-s − 4.94·25-s − 8.59·26-s − 27-s + 0.107·28-s + ⋯
L(s)  = 1  + 0.980·2-s − 0.577·3-s − 0.0392·4-s + 0.100·5-s − 0.565·6-s − 0.514·7-s − 1.01·8-s + 0.333·9-s + 0.0981·10-s − 0.547·11-s + 0.0226·12-s − 1.71·13-s − 0.504·14-s − 0.0578·15-s − 0.959·16-s − 0.242·17-s + 0.326·18-s + 0.0693·19-s − 0.00393·20-s + 0.297·21-s − 0.537·22-s + 0.740·23-s + 0.588·24-s − 0.989·25-s − 1.68·26-s − 0.192·27-s + 0.0202·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.298566736$
$L(\frac12)$  $\approx$  $1.298566736$
$L(\frac{3}{2})$   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 \( 1 - 1.38T + 2T^{2} \)
5 \( 1 - 0.223T + 5T^{2} \)
7 \( 1 + 1.36T + 7T^{2} \)
11 \( 1 + 1.81T + 11T^{2} \)
13 \( 1 + 6.19T + 13T^{2} \)
19 \( 1 - 0.302T + 19T^{2} \)
23 \( 1 - 3.55T + 23T^{2} \)
29 \( 1 - 5.25T + 29T^{2} \)
31 \( 1 - 2.62T + 31T^{2} \)
37 \( 1 - 9.66T + 37T^{2} \)
41 \( 1 - 4.75T + 41T^{2} \)
43 \( 1 + 0.704T + 43T^{2} \)
47 \( 1 + 3.93T + 47T^{2} \)
53 \( 1 - 3.80T + 53T^{2} \)
59 \( 1 - 9.63T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 - 7.08T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
83 \( 1 - 6.16T + 83T^{2} \)
89 \( 1 - 0.122T + 89T^{2} \)
97 \( 1 + 2.12T + 97T^{2} \)
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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

−8.371094754791568972482246197930, −7.51732650967694865509314555145, −6.74597909736839022284414283788, −6.06483865543705999338653444811, −5.31691132611823973132907996717, −4.75732848008696600657300935777, −4.10753334477870706209631584497, −2.98357748420927904812553572813, −2.37215011449485719833506266434, −0.54960396624243223261140845623, 0.54960396624243223261140845623, 2.37215011449485719833506266434, 2.98357748420927904812553572813, 4.10753334477870706209631584497, 4.75732848008696600657300935777, 5.31691132611823973132907996717, 6.06483865543705999338653444811, 6.74597909736839022284414283788, 7.51732650967694865509314555145, 8.371094754791568972482246197930

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