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  − 1.21·2-s + 1.67·3-s − 0.532·4-s − 5-s − 2.03·6-s − 7-s + 3.06·8-s − 0.179·9-s + 1.21·10-s − 5.20·11-s − 0.893·12-s + 4.41·13-s + 1.21·14-s − 1.67·15-s − 2.65·16-s − 0.830·17-s + 0.217·18-s − 7.19·19-s + 0.532·20-s − 1.67·21-s + 6.30·22-s − 1.03·23-s + 5.15·24-s + 25-s − 5.34·26-s − 5.33·27-s + 0.532·28-s + ⋯
L(s)  = 1  − 0.856·2-s + 0.969·3-s − 0.266·4-s − 0.447·5-s − 0.830·6-s − 0.377·7-s + 1.08·8-s − 0.0599·9-s + 0.383·10-s − 1.56·11-s − 0.258·12-s + 1.22·13-s + 0.323·14-s − 0.433·15-s − 0.662·16-s − 0.201·17-s + 0.0513·18-s − 1.65·19-s + 0.119·20-s − 0.366·21-s + 1.34·22-s − 0.215·23-s + 1.05·24-s + 0.200·25-s − 1.04·26-s − 1.02·27-s + 0.100·28-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  =  $(2,\ 8015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6104053170$
$L(\frac12)$  $\approx$  $0.6104053170$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + T \)
7 \( 1 + T \)
229 \( 1 - T \)
good2 \( 1 + 1.21T + 2T^{2} \)
3 \( 1 - 1.67T + 3T^{2} \)
11 \( 1 + 5.20T + 11T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
17 \( 1 + 0.830T + 17T^{2} \)
19 \( 1 + 7.19T + 19T^{2} \)
23 \( 1 + 1.03T + 23T^{2} \)
29 \( 1 + 6.71T + 29T^{2} \)
31 \( 1 - 0.399T + 31T^{2} \)
37 \( 1 - 6.46T + 37T^{2} \)
41 \( 1 - 6.34T + 41T^{2} \)
43 \( 1 + 1.63T + 43T^{2} \)
47 \( 1 + 5.69T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 14.5T + 59T^{2} \)
61 \( 1 - 8.62T + 61T^{2} \)
67 \( 1 + 2.55T + 67T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 - 6.22T + 73T^{2} \)
79 \( 1 - 8.09T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 + 5.80T + 89T^{2} \)
97 \( 1 + 8.15T + 97T^{2} \)
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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

−8.107333720175987098220527765185, −7.65747269367685036540135010368, −6.60289536960090079534769851964, −5.85764404303917410397032428981, −4.91992453564019730991868470347, −4.09335192501166766317382567095, −3.48457413864833873689458082718, −2.56260953677051624914504948138, −1.81344167820489148585881152642, −0.40487953511880449598322869889, 0.40487953511880449598322869889, 1.81344167820489148585881152642, 2.56260953677051624914504948138, 3.48457413864833873689458082718, 4.09335192501166766317382567095, 4.91992453564019730991868470347, 5.85764404303917410397032428981, 6.60289536960090079534769851964, 7.65747269367685036540135010368, 8.107333720175987098220527765185

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