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  − 1.42·2-s + 3-s + 0.0397·4-s − 3.43·5-s − 1.42·6-s + 0.0177·7-s + 2.79·8-s + 9-s + 4.90·10-s − 2.14·11-s + 0.0397·12-s − 1.57·13-s − 0.0253·14-s − 3.43·15-s − 4.07·16-s − 17-s − 1.42·18-s − 1.12·19-s − 0.136·20-s + 0.0177·21-s + 3.05·22-s + 2.11·23-s + 2.79·24-s + 6.80·25-s + 2.24·26-s + 27-s + 0.000705·28-s + ⋯
L(s)  = 1  − 1.00·2-s + 0.577·3-s + 0.0198·4-s − 1.53·5-s − 0.583·6-s + 0.00670·7-s + 0.989·8-s + 0.333·9-s + 1.55·10-s − 0.645·11-s + 0.0114·12-s − 0.436·13-s − 0.00677·14-s − 0.887·15-s − 1.01·16-s − 0.242·17-s − 0.336·18-s − 0.258·19-s − 0.0305·20-s + 0.00387·21-s + 0.651·22-s + 0.441·23-s + 0.571·24-s + 1.36·25-s + 0.440·26-s + 0.192·27-s + 0.000133·28-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
17 \( 1 + T \)
157 \( 1 - T \)
good2 \( 1 + 1.42T + 2T^{2} \)
5 \( 1 + 3.43T + 5T^{2} \)
7 \( 1 - 0.0177T + 7T^{2} \)
11 \( 1 + 2.14T + 11T^{2} \)
13 \( 1 + 1.57T + 13T^{2} \)
19 \( 1 + 1.12T + 19T^{2} \)
23 \( 1 - 2.11T + 23T^{2} \)
29 \( 1 - 7.22T + 29T^{2} \)
31 \( 1 + 2.53T + 31T^{2} \)
37 \( 1 - 5.42T + 37T^{2} \)
41 \( 1 - 9.59T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 4.78T + 53T^{2} \)
59 \( 1 - 1.42T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 + 5.47T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 3.38T + 79T^{2} \)
83 \( 1 - 8.56T + 83T^{2} \)
89 \( 1 + 8.56T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−7.922340078362409807871415990235, −7.09599275600880314251543124137, −6.54657702272778722132433444090, −5.04825697944317294491599454149, −4.64847427997886466716939075365, −3.85375202746429453776183599873, −3.09176357828743042022628497174, −2.15156628014833822998620856610, −0.921769575436422684557626593663, 0, 0.921769575436422684557626593663, 2.15156628014833822998620856610, 3.09176357828743042022628497174, 3.85375202746429453776183599873, 4.64847427997886466716939075365, 5.04825697944317294491599454149, 6.54657702272778722132433444090, 7.09599275600880314251543124137, 7.922340078362409807871415990235

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