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  + 1.99·2-s − 3-s + 1.97·4-s + 1.60·5-s − 1.99·6-s − 0.432·7-s − 0.0559·8-s + 9-s + 3.19·10-s + 1.01·11-s − 1.97·12-s − 6.62·13-s − 0.862·14-s − 1.60·15-s − 4.05·16-s − 17-s + 1.99·18-s + 1.52·19-s + 3.15·20-s + 0.432·21-s + 2.01·22-s + 4.61·23-s + 0.0559·24-s − 2.43·25-s − 13.2·26-s − 27-s − 0.853·28-s + ⋯
L(s)  = 1  + 1.40·2-s − 0.577·3-s + 0.985·4-s + 0.716·5-s − 0.813·6-s − 0.163·7-s − 0.0197·8-s + 0.333·9-s + 1.00·10-s + 0.305·11-s − 0.569·12-s − 1.83·13-s − 0.230·14-s − 0.413·15-s − 1.01·16-s − 0.242·17-s + 0.469·18-s + 0.349·19-s + 0.705·20-s + 0.0944·21-s + 0.429·22-s + 0.962·23-s + 0.0114·24-s − 0.487·25-s − 2.58·26-s − 0.192·27-s − 0.161·28-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 \( 1 - 1.99T + 2T^{2} \)
5 \( 1 - 1.60T + 5T^{2} \)
7 \( 1 + 0.432T + 7T^{2} \)
11 \( 1 - 1.01T + 11T^{2} \)
13 \( 1 + 6.62T + 13T^{2} \)
19 \( 1 - 1.52T + 19T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 - 1.72T + 31T^{2} \)
37 \( 1 + 2.02T + 37T^{2} \)
41 \( 1 + 0.00303T + 41T^{2} \)
43 \( 1 - 2.14T + 43T^{2} \)
47 \( 1 + 3.31T + 47T^{2} \)
53 \( 1 + 4.49T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 + 1.11T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + 7.62T + 71T^{2} \)
73 \( 1 - 8.54T + 73T^{2} \)
79 \( 1 - 4.23T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 - 9.19T + 89T^{2} \)
97 \( 1 - 0.643T + 97T^{2} \)
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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

−7.09883843523671599059579701036, −6.51928664814853930370557964819, −6.04045230640758047527755094897, −5.13018699027931906255052274439, −4.87657443053649270788425826079, −4.20069555457518994389141988448, −3.04400190206853359435078625686, −2.60428881086815979415618771575, −1.50440359373712454239419171791, 0, 1.50440359373712454239419171791, 2.60428881086815979415618771575, 3.04400190206853359435078625686, 4.20069555457518994389141988448, 4.87657443053649270788425826079, 5.13018699027931906255052274439, 6.04045230640758047527755094897, 6.51928664814853930370557964819, 7.09883843523671599059579701036

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