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  + 2.47·2-s − 3-s + 4.11·4-s + 2.50·5-s − 2.47·6-s − 1.59·7-s + 5.22·8-s + 9-s + 6.19·10-s − 4.88·11-s − 4.11·12-s − 3.74·13-s − 3.94·14-s − 2.50·15-s + 4.68·16-s − 17-s + 2.47·18-s − 5.54·19-s + 10.3·20-s + 1.59·21-s − 12.0·22-s + 5.99·23-s − 5.22·24-s + 1.28·25-s − 9.26·26-s − 27-s − 6.56·28-s + ⋯
L(s)  = 1  + 1.74·2-s − 0.577·3-s + 2.05·4-s + 1.12·5-s − 1.00·6-s − 0.603·7-s + 1.84·8-s + 0.333·9-s + 1.95·10-s − 1.47·11-s − 1.18·12-s − 1.03·13-s − 1.05·14-s − 0.647·15-s + 1.17·16-s − 0.242·17-s + 0.582·18-s − 1.27·19-s + 2.30·20-s + 0.348·21-s − 2.57·22-s + 1.25·23-s − 1.06·24-s + 0.256·25-s − 1.81·26-s − 0.192·27-s − 1.23·28-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 \( 1 - 2.47T + 2T^{2} \)
5 \( 1 - 2.50T + 5T^{2} \)
7 \( 1 + 1.59T + 7T^{2} \)
11 \( 1 + 4.88T + 11T^{2} \)
13 \( 1 + 3.74T + 13T^{2} \)
19 \( 1 + 5.54T + 19T^{2} \)
23 \( 1 - 5.99T + 23T^{2} \)
29 \( 1 + 3.31T + 29T^{2} \)
31 \( 1 + 2.69T + 31T^{2} \)
37 \( 1 - 6.21T + 37T^{2} \)
41 \( 1 - 1.59T + 41T^{2} \)
43 \( 1 - 8.43T + 43T^{2} \)
47 \( 1 - 2.21T + 47T^{2} \)
53 \( 1 + 13.5T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 6.89T + 61T^{2} \)
67 \( 1 + 3.64T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + 7.03T + 73T^{2} \)
79 \( 1 + 6.59T + 79T^{2} \)
83 \( 1 - 1.65T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + 5.29T + 97T^{2} \)
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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

−7.14946809226581882103729208059, −6.44880876811741940701880394964, −5.84997446066091798775610267151, −5.45354812662322587594873502778, −4.75284811965667916928345012613, −4.21976879061458279702608117688, −2.96583441845386795717794713333, −2.58136367234902296972448447653, −1.78045948726753788766286333170, 0, 1.78045948726753788766286333170, 2.58136367234902296972448447653, 2.96583441845386795717794713333, 4.21976879061458279702608117688, 4.75284811965667916928345012613, 5.45354812662322587594873502778, 5.84997446066091798775610267151, 6.44880876811741940701880394964, 7.14946809226581882103729208059

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