Properties

Degree 2
Conductor $ 5 \cdot 7 \cdot 229 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.20·2-s − 0.0912·3-s − 0.539·4-s − 5-s − 0.110·6-s − 7-s − 3.06·8-s − 2.99·9-s − 1.20·10-s + 1.74·11-s + 0.0491·12-s − 0.184·13-s − 1.20·14-s + 0.0912·15-s − 2.63·16-s − 2.26·17-s − 3.61·18-s + 2.30·19-s + 0.539·20-s + 0.0912·21-s + 2.10·22-s − 5.73·23-s + 0.279·24-s + 25-s − 0.222·26-s + 0.546·27-s + 0.539·28-s + ⋯
L(s)  = 1  + 0.854·2-s − 0.0526·3-s − 0.269·4-s − 0.447·5-s − 0.0450·6-s − 0.377·7-s − 1.08·8-s − 0.997·9-s − 0.382·10-s + 0.526·11-s + 0.0141·12-s − 0.0511·13-s − 0.323·14-s + 0.0235·15-s − 0.657·16-s − 0.549·17-s − 0.852·18-s + 0.528·19-s + 0.120·20-s + 0.0199·21-s + 0.449·22-s − 1.19·23-s + 0.0571·24-s + 0.200·25-s − 0.0437·26-s + 0.105·27-s + 0.101·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8015\)    =    \(5 \cdot 7 \cdot 229\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8015} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.110922270$
$L(\frac12)$  $\approx$  $1.110922270$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;7,\;229\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;7,\;229\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 + T \)
229 \( 1 - T \)
good2 \( 1 - 1.20T + 2T^{2} \)
3 \( 1 + 0.0912T + 3T^{2} \)
11 \( 1 - 1.74T + 11T^{2} \)
13 \( 1 + 0.184T + 13T^{2} \)
17 \( 1 + 2.26T + 17T^{2} \)
19 \( 1 - 2.30T + 19T^{2} \)
23 \( 1 + 5.73T + 23T^{2} \)
29 \( 1 - 5.32T + 29T^{2} \)
31 \( 1 + 3.10T + 31T^{2} \)
37 \( 1 + 5.98T + 37T^{2} \)
41 \( 1 - 8.60T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 7.18T + 53T^{2} \)
59 \( 1 + 14.8T + 59T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 - 7.58T + 73T^{2} \)
79 \( 1 + 9.71T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−7.894696153159598202377956635920, −6.93585296725127335721136141331, −6.22085657467902485706180438708, −5.78904321873415752284085555918, −4.87372910673433377139397365448, −4.39682823328849218598025788864, −3.38597834998770349031193852577, −3.16733269863130495013769265328, −1.96763872541190108166900670607, −0.43874180806373513893037766785, 0.43874180806373513893037766785, 1.96763872541190108166900670607, 3.16733269863130495013769265328, 3.38597834998770349031193852577, 4.39682823328849218598025788864, 4.87372910673433377139397365448, 5.78904321873415752284085555918, 6.22085657467902485706180438708, 6.93585296725127335721136141331, 7.894696153159598202377956635920

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