Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.557·2-s − 3-s − 1.68·4-s + 2.57·5-s + 0.557·6-s + 2.28·7-s + 2.05·8-s + 9-s − 1.43·10-s − 2.42·11-s + 1.68·12-s + 5.98·13-s − 1.27·14-s − 2.57·15-s + 2.23·16-s − 17-s − 0.557·18-s + 4.65·19-s − 4.34·20-s − 2.28·21-s + 1.35·22-s + 0.860·23-s − 2.05·24-s + 1.60·25-s − 3.33·26-s − 27-s − 3.85·28-s + ⋯
L(s)  = 1  − 0.394·2-s − 0.577·3-s − 0.844·4-s + 1.14·5-s + 0.227·6-s + 0.862·7-s + 0.727·8-s + 0.333·9-s − 0.453·10-s − 0.731·11-s + 0.487·12-s + 1.66·13-s − 0.339·14-s − 0.663·15-s + 0.557·16-s − 0.242·17-s − 0.131·18-s + 1.06·19-s − 0.970·20-s − 0.497·21-s + 0.288·22-s + 0.179·23-s − 0.419·24-s + 0.321·25-s − 0.654·26-s − 0.192·27-s − 0.728·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 + 0.557T + 2T^{2} \)
5 \( 1 - 2.57T + 5T^{2} \)
7 \( 1 - 2.28T + 7T^{2} \)
11 \( 1 + 2.42T + 11T^{2} \)
13 \( 1 - 5.98T + 13T^{2} \)
19 \( 1 - 4.65T + 19T^{2} \)
23 \( 1 - 0.860T + 23T^{2} \)
29 \( 1 + 5.60T + 29T^{2} \)
31 \( 1 + 7.14T + 31T^{2} \)
37 \( 1 + 7.55T + 37T^{2} \)
41 \( 1 - 7.64T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 1.24T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 - 0.360T + 67T^{2} \)
71 \( 1 + 6.84T + 71T^{2} \)
73 \( 1 + 6.78T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 7.63T + 83T^{2} \)
89 \( 1 + 5.84T + 89T^{2} \)
97 \( 1 + 2.78T + 97T^{2} \)
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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

−7.61223791178052122428046975417, −6.80267691463296334270090568116, −5.79327990205477397519297587279, −5.48773915647963809182427807172, −4.91665083045255075597534476102, −4.01026217320550899583129034896, −3.16270546310824184808046678414, −1.62698570161277583791054356336, −1.47663687812598018073775719880, 0, 1.47663687812598018073775719880, 1.62698570161277583791054356336, 3.16270546310824184808046678414, 4.01026217320550899583129034896, 4.91665083045255075597534476102, 5.48773915647963809182427807172, 5.79327990205477397519297587279, 6.80267691463296334270090568116, 7.61223791178052122428046975417

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