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  − 0.498·2-s + 3-s − 1.75·4-s + 0.0200·5-s − 0.498·6-s − 2.67·7-s + 1.87·8-s + 9-s − 0.0100·10-s + 0.400·11-s − 1.75·12-s + 1.83·13-s + 1.33·14-s + 0.0200·15-s + 2.56·16-s − 17-s − 0.498·18-s − 2.84·19-s − 0.0351·20-s − 2.67·21-s − 0.199·22-s + 6.13·23-s + 1.87·24-s − 4.99·25-s − 0.916·26-s + 27-s + 4.67·28-s + ⋯
L(s)  = 1  − 0.352·2-s + 0.577·3-s − 0.875·4-s + 0.00896·5-s − 0.203·6-s − 1.00·7-s + 0.661·8-s + 0.333·9-s − 0.00316·10-s + 0.120·11-s − 0.505·12-s + 0.509·13-s + 0.356·14-s + 0.00517·15-s + 0.642·16-s − 0.242·17-s − 0.117·18-s − 0.652·19-s − 0.00785·20-s − 0.582·21-s − 0.0425·22-s + 1.27·23-s + 0.382·24-s − 0.999·25-s − 0.179·26-s + 0.192·27-s + 0.883·28-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 \( 1 + 0.498T + 2T^{2} \)
5 \( 1 - 0.0200T + 5T^{2} \)
7 \( 1 + 2.67T + 7T^{2} \)
11 \( 1 - 0.400T + 11T^{2} \)
13 \( 1 - 1.83T + 13T^{2} \)
19 \( 1 + 2.84T + 19T^{2} \)
23 \( 1 - 6.13T + 23T^{2} \)
29 \( 1 + 3.39T + 29T^{2} \)
31 \( 1 + 4.05T + 31T^{2} \)
37 \( 1 - 6.09T + 37T^{2} \)
41 \( 1 - 8.26T + 41T^{2} \)
43 \( 1 + 0.679T + 43T^{2} \)
47 \( 1 + 5.43T + 47T^{2} \)
53 \( 1 - 9.14T + 53T^{2} \)
59 \( 1 + 9.11T + 59T^{2} \)
61 \( 1 + 5.01T + 61T^{2} \)
67 \( 1 - 4.00T + 67T^{2} \)
71 \( 1 - 0.282T + 71T^{2} \)
73 \( 1 - 3.43T + 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 - 8.64T + 83T^{2} \)
89 \( 1 - 4.48T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−7.68813579742520110640655136949, −6.88124600421850433571212995677, −6.16888163261313489318828926105, −5.41458579800867545575326971018, −4.47430018383654312410761396680, −3.84527025265927859447146909286, −3.23580117208709954671382887438, −2.23271758414818087893352138894, −1.12592063727972048621488811972, 0, 1.12592063727972048621488811972, 2.23271758414818087893352138894, 3.23580117208709954671382887438, 3.84527025265927859447146909286, 4.47430018383654312410761396680, 5.41458579800867545575326971018, 6.16888163261313489318828926105, 6.88124600421850433571212995677, 7.68813579742520110640655136949

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