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  − 2.39·2-s − 3-s + 3.72·4-s + 0.325·5-s + 2.39·6-s − 0.363·7-s − 4.12·8-s + 9-s − 0.777·10-s − 0.861·11-s − 3.72·12-s − 0.164·13-s + 0.868·14-s − 0.325·15-s + 2.41·16-s − 17-s − 2.39·18-s − 8.07·19-s + 1.21·20-s + 0.363·21-s + 2.05·22-s + 3.99·23-s + 4.12·24-s − 4.89·25-s + 0.392·26-s − 27-s − 1.35·28-s + ⋯
L(s)  = 1  − 1.69·2-s − 0.577·3-s + 1.86·4-s + 0.145·5-s + 0.976·6-s − 0.137·7-s − 1.45·8-s + 0.333·9-s − 0.245·10-s − 0.259·11-s − 1.07·12-s − 0.0455·13-s + 0.232·14-s − 0.0839·15-s + 0.603·16-s − 0.242·17-s − 0.563·18-s − 1.85·19-s + 0.270·20-s + 0.0792·21-s + 0.439·22-s + 0.833·23-s + 0.841·24-s − 0.978·25-s + 0.0769·26-s − 0.192·27-s − 0.255·28-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 + 2.39T + 2T^{2} \)
5 \( 1 - 0.325T + 5T^{2} \)
7 \( 1 + 0.363T + 7T^{2} \)
11 \( 1 + 0.861T + 11T^{2} \)
13 \( 1 + 0.164T + 13T^{2} \)
19 \( 1 + 8.07T + 19T^{2} \)
23 \( 1 - 3.99T + 23T^{2} \)
29 \( 1 + 3.43T + 29T^{2} \)
31 \( 1 - 5.64T + 31T^{2} \)
37 \( 1 - 5.74T + 37T^{2} \)
41 \( 1 + 3.17T + 41T^{2} \)
43 \( 1 - 8.78T + 43T^{2} \)
47 \( 1 - 4.83T + 47T^{2} \)
53 \( 1 + 4.72T + 53T^{2} \)
59 \( 1 - 5.82T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 5.13T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 - 17.9T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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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

−7.72647674046584063887432149581, −6.80732487566116751250415261391, −6.45039282279772654743456168869, −5.70349217216023980840267319407, −4.71489447521073358905397040470, −3.90654106281295940952722034004, −2.58573161718171439660995881078, −2.00851800100680957001713587430, −0.927247716246292933203885636735, 0, 0.927247716246292933203885636735, 2.00851800100680957001713587430, 2.58573161718171439660995881078, 3.90654106281295940952722034004, 4.71489447521073358905397040470, 5.70349217216023980840267319407, 6.45039282279772654743456168869, 6.80732487566116751250415261391, 7.72647674046584063887432149581

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