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  + 2.22·2-s + 3-s + 2.94·4-s − 1.48·5-s + 2.22·6-s + 3.09·7-s + 2.10·8-s + 9-s − 3.30·10-s − 5.38·11-s + 2.94·12-s − 2.35·13-s + 6.88·14-s − 1.48·15-s − 1.21·16-s − 17-s + 2.22·18-s − 3.03·19-s − 4.37·20-s + 3.09·21-s − 11.9·22-s − 4.35·23-s + 2.10·24-s − 2.79·25-s − 5.24·26-s + 27-s + 9.12·28-s + ⋯
L(s)  = 1  + 1.57·2-s + 0.577·3-s + 1.47·4-s − 0.664·5-s + 0.907·6-s + 1.17·7-s + 0.744·8-s + 0.333·9-s − 1.04·10-s − 1.62·11-s + 0.850·12-s − 0.654·13-s + 1.84·14-s − 0.383·15-s − 0.302·16-s − 0.242·17-s + 0.524·18-s − 0.695·19-s − 0.978·20-s + 0.675·21-s − 2.55·22-s − 0.909·23-s + 0.429·24-s − 0.558·25-s − 1.02·26-s + 0.192·27-s + 1.72·28-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 \( 1 - 2.22T + 2T^{2} \)
5 \( 1 + 1.48T + 5T^{2} \)
7 \( 1 - 3.09T + 7T^{2} \)
11 \( 1 + 5.38T + 11T^{2} \)
13 \( 1 + 2.35T + 13T^{2} \)
19 \( 1 + 3.03T + 19T^{2} \)
23 \( 1 + 4.35T + 23T^{2} \)
29 \( 1 - 7.85T + 29T^{2} \)
31 \( 1 + 9.03T + 31T^{2} \)
37 \( 1 - 9.47T + 37T^{2} \)
41 \( 1 + 8.20T + 41T^{2} \)
43 \( 1 - 2.13T + 43T^{2} \)
47 \( 1 - 2.22T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 5.67T + 59T^{2} \)
61 \( 1 - 5.05T + 61T^{2} \)
67 \( 1 - 1.62T + 67T^{2} \)
71 \( 1 + 4.40T + 71T^{2} \)
73 \( 1 + 4.38T + 73T^{2} \)
79 \( 1 + 1.04T + 79T^{2} \)
83 \( 1 - 3.94T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 5.65T + 97T^{2} \)
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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

−7.67054678746356917474954105467, −6.70926038043406413677307202751, −5.85617907091558210719561648325, −5.10429946894714638344144864905, −4.62496763512114298282783162723, −4.09982308387517539654251302126, −3.21110155315464740337470675064, −2.45588065370265671167006901333, −1.88227762251093791100248283034, 0, 1.88227762251093791100248283034, 2.45588065370265671167006901333, 3.21110155315464740337470675064, 4.09982308387517539654251302126, 4.62496763512114298282783162723, 5.10429946894714638344144864905, 5.85617907091558210719561648325, 6.70926038043406413677307202751, 7.67054678746356917474954105467

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