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  − 2.00·2-s + 3-s + 2.01·4-s + 2.86·5-s − 2.00·6-s − 1.37·7-s − 0.0389·8-s + 9-s − 5.74·10-s + 4.45·11-s + 2.01·12-s − 0.546·13-s + 2.76·14-s + 2.86·15-s − 3.96·16-s − 17-s − 2.00·18-s − 5.48·19-s + 5.78·20-s − 1.37·21-s − 8.93·22-s − 5.85·23-s − 0.0389·24-s + 3.21·25-s + 1.09·26-s + 27-s − 2.78·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 1.00·4-s + 1.28·5-s − 0.818·6-s − 0.520·7-s − 0.0137·8-s + 0.333·9-s − 1.81·10-s + 1.34·11-s + 0.582·12-s − 0.151·13-s + 0.737·14-s + 0.740·15-s − 0.990·16-s − 0.242·17-s − 0.472·18-s − 1.25·19-s + 1.29·20-s − 0.300·21-s − 1.90·22-s − 1.22·23-s − 0.00795·24-s + 0.643·25-s + 0.214·26-s + 0.192·27-s − 0.525·28-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 \( 1 + 2.00T + 2T^{2} \)
5 \( 1 - 2.86T + 5T^{2} \)
7 \( 1 + 1.37T + 7T^{2} \)
11 \( 1 - 4.45T + 11T^{2} \)
13 \( 1 + 0.546T + 13T^{2} \)
19 \( 1 + 5.48T + 19T^{2} \)
23 \( 1 + 5.85T + 23T^{2} \)
29 \( 1 - 1.43T + 29T^{2} \)
31 \( 1 + 6.52T + 31T^{2} \)
37 \( 1 - 2.12T + 37T^{2} \)
41 \( 1 - 4.65T + 41T^{2} \)
43 \( 1 + 3.60T + 43T^{2} \)
47 \( 1 + 3.40T + 47T^{2} \)
53 \( 1 + 3.06T + 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 + 4.18T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 - 1.13T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + 4.80T + 79T^{2} \)
83 \( 1 - 2.37T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−7.63852040680868516918774480569, −6.83795356367686902995138959499, −6.39398530972969482002793546909, −5.75138129119930579241219211172, −4.50895032482123088147454225295, −3.85470456381661045498580560610, −2.67708065840840844999346487314, −1.90396710847812961701037034559, −1.42307826119273183385881922342, 0, 1.42307826119273183385881922342, 1.90396710847812961701037034559, 2.67708065840840844999346487314, 3.85470456381661045498580560610, 4.50895032482123088147454225295, 5.75138129119930579241219211172, 6.39398530972969482002793546909, 6.83795356367686902995138959499, 7.63852040680868516918774480569

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