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  − 1.59·2-s + 3-s + 0.554·4-s + 1.16·5-s − 1.59·6-s + 1.82·7-s + 2.30·8-s + 9-s − 1.87·10-s − 1.19·11-s + 0.554·12-s + 0.914·13-s − 2.91·14-s + 1.16·15-s − 4.80·16-s − 17-s − 1.59·18-s + 7.92·19-s + 0.649·20-s + 1.82·21-s + 1.90·22-s − 4.14·23-s + 2.30·24-s − 3.63·25-s − 1.46·26-s + 27-s + 1.01·28-s + ⋯
L(s)  = 1  − 1.13·2-s + 0.577·3-s + 0.277·4-s + 0.523·5-s − 0.652·6-s + 0.689·7-s + 0.816·8-s + 0.333·9-s − 0.591·10-s − 0.359·11-s + 0.160·12-s + 0.253·13-s − 0.779·14-s + 0.302·15-s − 1.20·16-s − 0.242·17-s − 0.376·18-s + 1.81·19-s + 0.145·20-s + 0.398·21-s + 0.406·22-s − 0.863·23-s + 0.471·24-s − 0.726·25-s − 0.286·26-s + 0.192·27-s + 0.191·28-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 \( 1 + 1.59T + 2T^{2} \)
5 \( 1 - 1.16T + 5T^{2} \)
7 \( 1 - 1.82T + 7T^{2} \)
11 \( 1 + 1.19T + 11T^{2} \)
13 \( 1 - 0.914T + 13T^{2} \)
19 \( 1 - 7.92T + 19T^{2} \)
23 \( 1 + 4.14T + 23T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 + 6.18T + 31T^{2} \)
37 \( 1 - 4.92T + 37T^{2} \)
41 \( 1 + 6.06T + 41T^{2} \)
43 \( 1 - 2.66T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 3.73T + 59T^{2} \)
61 \( 1 + 7.41T + 61T^{2} \)
67 \( 1 + 6.67T + 67T^{2} \)
71 \( 1 + 0.615T + 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 4.29T + 83T^{2} \)
89 \( 1 + 1.40T + 89T^{2} \)
97 \( 1 - 7.58T + 97T^{2} \)
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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

−7.82579192629778488556571337939, −7.20734431404644540539516877519, −6.20917001866798755697507621879, −5.37201177367462044249059731995, −4.72477714052039809506800710819, −3.82204816318301849383481679884, −2.91189149185650464659641037844, −1.78276426056591432285892031588, −1.46153903228169328189680153414, 0, 1.46153903228169328189680153414, 1.78276426056591432285892031588, 2.91189149185650464659641037844, 3.82204816318301849383481679884, 4.72477714052039809506800710819, 5.37201177367462044249059731995, 6.20917001866798755697507621879, 7.20734431404644540539516877519, 7.82579192629778488556571337939

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