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  − 2.78·2-s − 3-s + 5.73·4-s + 2.04·5-s + 2.78·6-s − 0.793·7-s − 10.3·8-s + 9-s − 5.68·10-s + 1.06·11-s − 5.73·12-s + 3.98·13-s + 2.20·14-s − 2.04·15-s + 17.4·16-s − 17-s − 2.78·18-s − 5.59·19-s + 11.7·20-s + 0.793·21-s − 2.95·22-s + 3.28·23-s + 10.3·24-s − 0.821·25-s − 11.0·26-s − 27-s − 4.54·28-s + ⋯
L(s)  = 1  − 1.96·2-s − 0.577·3-s + 2.86·4-s + 0.914·5-s + 1.13·6-s − 0.299·7-s − 3.67·8-s + 0.333·9-s − 1.79·10-s + 0.320·11-s − 1.65·12-s + 1.10·13-s + 0.589·14-s − 0.527·15-s + 4.35·16-s − 0.242·17-s − 0.655·18-s − 1.28·19-s + 2.62·20-s + 0.173·21-s − 0.629·22-s + 0.683·23-s + 2.12·24-s − 0.164·25-s − 2.17·26-s − 0.192·27-s − 0.859·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

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 \( 1 + 2.78T + 2T^{2} \)
5 \( 1 - 2.04T + 5T^{2} \)
7 \( 1 + 0.793T + 7T^{2} \)
11 \( 1 - 1.06T + 11T^{2} \)
13 \( 1 - 3.98T + 13T^{2} \)
19 \( 1 + 5.59T + 19T^{2} \)
23 \( 1 - 3.28T + 23T^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
31 \( 1 - 1.66T + 31T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 - 0.684T + 41T^{2} \)
43 \( 1 + 1.07T + 43T^{2} \)
47 \( 1 - 1.11T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 3.58T + 59T^{2} \)
61 \( 1 + 2.01T + 61T^{2} \)
67 \( 1 + 6.63T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 8.90T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 + 3.21T + 89T^{2} \)
97 \( 1 - 7.38T + 97T^{2} \)
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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

−7.59432705899267966133386159468, −6.60333597108920502874407304770, −6.48908674391351952039896216248, −5.94440682122075342626579343576, −4.90928492422340332902305727097, −3.58183529213442800522938556459, −2.66766661712780976162528244276, −1.76480668533037768579834949589, −1.16325436394722654962578535137, 0, 1.16325436394722654962578535137, 1.76480668533037768579834949589, 2.66766661712780976162528244276, 3.58183529213442800522938556459, 4.90928492422340332902305727097, 5.94440682122075342626579343576, 6.48908674391351952039896216248, 6.60333597108920502874407304770, 7.59432705899267966133386159468

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