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  + 1.82·2-s − 3-s + 1.33·4-s − 0.167·5-s − 1.82·6-s + 3.32·7-s − 1.21·8-s + 9-s − 0.306·10-s − 0.790·11-s − 1.33·12-s − 2.24·13-s + 6.06·14-s + 0.167·15-s − 4.88·16-s − 17-s + 1.82·18-s + 7.55·19-s − 0.223·20-s − 3.32·21-s − 1.44·22-s + 0.417·23-s + 1.21·24-s − 4.97·25-s − 4.10·26-s − 27-s + 4.43·28-s + ⋯
L(s)  = 1  + 1.29·2-s − 0.577·3-s + 0.667·4-s − 0.0749·5-s − 0.745·6-s + 1.25·7-s − 0.429·8-s + 0.333·9-s − 0.0967·10-s − 0.238·11-s − 0.385·12-s − 0.622·13-s + 1.62·14-s + 0.0432·15-s − 1.22·16-s − 0.242·17-s + 0.430·18-s + 1.73·19-s − 0.0500·20-s − 0.725·21-s − 0.307·22-s + 0.0869·23-s + 0.248·24-s − 0.994·25-s − 0.804·26-s − 0.192·27-s + 0.838·28-s + ⋯

Functional equation

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

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 \( 1 - 1.82T + 2T^{2} \)
5 \( 1 + 0.167T + 5T^{2} \)
7 \( 1 - 3.32T + 7T^{2} \)
11 \( 1 + 0.790T + 11T^{2} \)
13 \( 1 + 2.24T + 13T^{2} \)
19 \( 1 - 7.55T + 19T^{2} \)
23 \( 1 - 0.417T + 23T^{2} \)
29 \( 1 + 2.98T + 29T^{2} \)
31 \( 1 + 8.95T + 31T^{2} \)
37 \( 1 + 1.22T + 37T^{2} \)
41 \( 1 + 9.37T + 41T^{2} \)
43 \( 1 + 2.86T + 43T^{2} \)
47 \( 1 - 4.19T + 47T^{2} \)
53 \( 1 - 3.33T + 53T^{2} \)
59 \( 1 - 5.24T + 59T^{2} \)
61 \( 1 + 4.29T + 61T^{2} \)
67 \( 1 + 2.10T + 67T^{2} \)
71 \( 1 - 3.75T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 6.26T + 83T^{2} \)
89 \( 1 - 5.46T + 89T^{2} \)
97 \( 1 - 1.46T + 97T^{2} \)
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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

−7.35257651054962254482018292666, −6.70329398081476814829355908319, −5.59003465569957732153845319381, −5.37660300385486235073076261391, −4.85781429862191544060951828164, −4.03646927264671822961371043803, −3.40171930842664545685518598541, −2.34047321807577543288592578125, −1.48482713008024727665758816509, 0, 1.48482713008024727665758816509, 2.34047321807577543288592578125, 3.40171930842664545685518598541, 4.03646927264671822961371043803, 4.85781429862191544060951828164, 5.37660300385486235073076261391, 5.59003465569957732153845319381, 6.70329398081476814829355908319, 7.35257651054962254482018292666

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