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  − 0.756·2-s + 3-s − 1.42·4-s − 3.23·5-s − 0.756·6-s + 4.69·7-s + 2.59·8-s + 9-s + 2.44·10-s + 3.89·11-s − 1.42·12-s + 0.979·13-s − 3.55·14-s − 3.23·15-s + 0.896·16-s − 17-s − 0.756·18-s − 4.21·19-s + 4.62·20-s + 4.69·21-s − 2.94·22-s − 3.94·23-s + 2.59·24-s + 5.48·25-s − 0.740·26-s + 27-s − 6.70·28-s + ⋯
L(s)  = 1  − 0.534·2-s + 0.577·3-s − 0.714·4-s − 1.44·5-s − 0.308·6-s + 1.77·7-s + 0.916·8-s + 0.333·9-s + 0.774·10-s + 1.17·11-s − 0.412·12-s + 0.271·13-s − 0.949·14-s − 0.836·15-s + 0.224·16-s − 0.242·17-s − 0.178·18-s − 0.965·19-s + 1.03·20-s + 1.02·21-s − 0.627·22-s − 0.821·23-s + 0.529·24-s + 1.09·25-s − 0.145·26-s + 0.192·27-s − 1.26·28-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 \( 1 + 0.756T + 2T^{2} \)
5 \( 1 + 3.23T + 5T^{2} \)
7 \( 1 - 4.69T + 7T^{2} \)
11 \( 1 - 3.89T + 11T^{2} \)
13 \( 1 - 0.979T + 13T^{2} \)
19 \( 1 + 4.21T + 19T^{2} \)
23 \( 1 + 3.94T + 23T^{2} \)
29 \( 1 - 5.35T + 29T^{2} \)
31 \( 1 + 3.41T + 31T^{2} \)
37 \( 1 + 9.92T + 37T^{2} \)
41 \( 1 + 2.77T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 + 7.88T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + 8.38T + 59T^{2} \)
61 \( 1 + 2.01T + 61T^{2} \)
67 \( 1 + 9.85T + 67T^{2} \)
71 \( 1 - 3.35T + 71T^{2} \)
73 \( 1 + 4.43T + 73T^{2} \)
79 \( 1 + 7.68T + 79T^{2} \)
83 \( 1 - 1.36T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + 9.58T + 97T^{2} \)
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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

−7.79296982042053117648115109120, −7.20558915168012075520108705683, −6.26038099920757232991920047212, −5.02458352476588338149422309919, −4.43031414749783426410735512279, −4.09990331692440138100891825830, −3.34452694627572720839674722413, −1.86690190223271389175191503676, −1.28862757600481299195377345374, 0, 1.28862757600481299195377345374, 1.86690190223271389175191503676, 3.34452694627572720839674722413, 4.09990331692440138100891825830, 4.43031414749783426410735512279, 5.02458352476588338149422309919, 6.26038099920757232991920047212, 7.20558915168012075520108705683, 7.79296982042053117648115109120

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