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  + 2.05·2-s + 3-s + 2.21·4-s + 2.38·5-s + 2.05·6-s − 5.11·7-s + 0.447·8-s + 9-s + 4.90·10-s + 1.96·11-s + 2.21·12-s − 2.07·13-s − 10.4·14-s + 2.38·15-s − 3.51·16-s − 17-s + 2.05·18-s − 2.46·19-s + 5.29·20-s − 5.11·21-s + 4.03·22-s − 8.81·23-s + 0.447·24-s + 0.695·25-s − 4.26·26-s + 27-s − 11.3·28-s + ⋯
L(s)  = 1  + 1.45·2-s + 0.577·3-s + 1.10·4-s + 1.06·5-s + 0.838·6-s − 1.93·7-s + 0.158·8-s + 0.333·9-s + 1.54·10-s + 0.591·11-s + 0.640·12-s − 0.576·13-s − 2.80·14-s + 0.616·15-s − 0.879·16-s − 0.242·17-s + 0.484·18-s − 0.565·19-s + 1.18·20-s − 1.11·21-s + 0.859·22-s − 1.83·23-s + 0.0912·24-s + 0.139·25-s − 0.837·26-s + 0.192·27-s − 2.14·28-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 \( 1 - 2.05T + 2T^{2} \)
5 \( 1 - 2.38T + 5T^{2} \)
7 \( 1 + 5.11T + 7T^{2} \)
11 \( 1 - 1.96T + 11T^{2} \)
13 \( 1 + 2.07T + 13T^{2} \)
19 \( 1 + 2.46T + 19T^{2} \)
23 \( 1 + 8.81T + 23T^{2} \)
29 \( 1 - 5.45T + 29T^{2} \)
31 \( 1 - 5.62T + 31T^{2} \)
37 \( 1 + 8.11T + 37T^{2} \)
41 \( 1 + 0.208T + 41T^{2} \)
43 \( 1 + 5.51T + 43T^{2} \)
47 \( 1 - 7.50T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 5.74T + 59T^{2} \)
61 \( 1 + 0.000881T + 61T^{2} \)
67 \( 1 + 5.07T + 67T^{2} \)
71 \( 1 - 7.80T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 - 2.62T + 79T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 - 3.16T + 97T^{2} \)
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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

−7.01285387119677244394454641486, −6.44866703044705940439447064443, −6.19952539477518182347000806223, −5.48113604129363828722286489516, −4.48954997527002069608665006755, −3.90506798942182058137160987955, −3.13999514869538994849199755068, −2.59268146538593279754196843955, −1.82211216960878127383327064056, 0, 1.82211216960878127383327064056, 2.59268146538593279754196843955, 3.13999514869538994849199755068, 3.90506798942182058137160987955, 4.48954997527002069608665006755, 5.48113604129363828722286489516, 6.19952539477518182347000806223, 6.44866703044705940439447064443, 7.01285387119677244394454641486

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