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  + 2.47·2-s − 3-s + 4.12·4-s + 3.86·5-s − 2.47·6-s + 4.55·7-s + 5.25·8-s + 9-s + 9.57·10-s + 1.38·11-s − 4.12·12-s − 5.68·13-s + 11.2·14-s − 3.86·15-s + 4.74·16-s − 17-s + 2.47·18-s + 5.64·19-s + 15.9·20-s − 4.55·21-s + 3.43·22-s − 9.45·23-s − 5.25·24-s + 9.96·25-s − 14.0·26-s − 27-s + 18.7·28-s + ⋯
L(s)  = 1  + 1.74·2-s − 0.577·3-s + 2.06·4-s + 1.73·5-s − 1.01·6-s + 1.71·7-s + 1.85·8-s + 0.333·9-s + 3.02·10-s + 0.418·11-s − 1.18·12-s − 1.57·13-s + 3.00·14-s − 0.998·15-s + 1.18·16-s − 0.242·17-s + 0.583·18-s + 1.29·19-s + 3.56·20-s − 0.992·21-s + 0.731·22-s − 1.97·23-s − 1.07·24-s + 1.99·25-s − 2.76·26-s − 0.192·27-s + 3.54·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.602217553$
$L(\frac12)$  $\approx$  $7.602217553$
$L(\frac{3}{2})$   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 \( 1 - 2.47T + 2T^{2} \)
5 \( 1 - 3.86T + 5T^{2} \)
7 \( 1 - 4.55T + 7T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
13 \( 1 + 5.68T + 13T^{2} \)
19 \( 1 - 5.64T + 19T^{2} \)
23 \( 1 + 9.45T + 23T^{2} \)
29 \( 1 + 4.69T + 29T^{2} \)
31 \( 1 - 5.97T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 + 0.951T + 41T^{2} \)
43 \( 1 + 6.02T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 + 8.40T + 59T^{2} \)
61 \( 1 - 3.16T + 61T^{2} \)
67 \( 1 - 0.624T + 67T^{2} \)
71 \( 1 - 2.42T + 71T^{2} \)
73 \( 1 + 2.93T + 73T^{2} \)
83 \( 1 + 2.89T + 83T^{2} \)
89 \( 1 + 7.26T + 89T^{2} \)
97 \( 1 + 8.31T + 97T^{2} \)
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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

−8.114546992420641012423306334734, −7.44395319227205751776430306560, −6.52939221652183264181607378708, −5.95075894325788752261581622190, −5.31569098211297854325765351986, −4.83941500350516423379443608061, −4.32025871832691055247464910195, −2.91927873885727356903917789630, −2.02274190217947888200020122819, −1.56954258149878046084104151450, 1.56954258149878046084104151450, 2.02274190217947888200020122819, 2.91927873885727356903917789630, 4.32025871832691055247464910195, 4.83941500350516423379443608061, 5.31569098211297854325765351986, 5.95075894325788752261581622190, 6.52939221652183264181607378708, 7.44395319227205751776430306560, 8.114546992420641012423306334734

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