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  − 2.17·2-s − 3-s + 2.74·4-s − 2.11·5-s + 2.17·6-s + 2.69·7-s − 1.63·8-s + 9-s + 4.60·10-s + 3.86·11-s − 2.74·12-s − 2.95·13-s − 5.87·14-s + 2.11·15-s − 1.94·16-s − 17-s − 2.17·18-s + 2.38·19-s − 5.80·20-s − 2.69·21-s − 8.42·22-s + 6.54·23-s + 1.63·24-s − 0.539·25-s + 6.44·26-s − 27-s + 7.41·28-s + ⋯
L(s)  = 1  − 1.54·2-s − 0.577·3-s + 1.37·4-s − 0.944·5-s + 0.889·6-s + 1.01·7-s − 0.576·8-s + 0.333·9-s + 1.45·10-s + 1.16·11-s − 0.793·12-s − 0.820·13-s − 1.57·14-s + 0.545·15-s − 0.485·16-s − 0.242·17-s − 0.513·18-s + 0.546·19-s − 1.29·20-s − 0.588·21-s − 1.79·22-s + 1.36·23-s + 0.332·24-s − 0.107·25-s + 1.26·26-s − 0.192·27-s + 1.40·28-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 \( 1 + 2.17T + 2T^{2} \)
5 \( 1 + 2.11T + 5T^{2} \)
7 \( 1 - 2.69T + 7T^{2} \)
11 \( 1 - 3.86T + 11T^{2} \)
13 \( 1 + 2.95T + 13T^{2} \)
19 \( 1 - 2.38T + 19T^{2} \)
23 \( 1 - 6.54T + 23T^{2} \)
29 \( 1 + 0.373T + 29T^{2} \)
31 \( 1 - 0.0561T + 31T^{2} \)
37 \( 1 - 2.78T + 37T^{2} \)
41 \( 1 + 4.44T + 41T^{2} \)
43 \( 1 + 8.21T + 43T^{2} \)
47 \( 1 - 9.32T + 47T^{2} \)
53 \( 1 - 4.89T + 53T^{2} \)
59 \( 1 - 8.24T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 3.51T + 67T^{2} \)
71 \( 1 + 12.0T + 71T^{2} \)
73 \( 1 + 3.35T + 73T^{2} \)
79 \( 1 + 9.02T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 4.41T + 89T^{2} \)
97 \( 1 - 17.3T + 97T^{2} \)
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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

−7.44335916453311745316726202381, −7.21063005497497898879488129951, −6.48289337539386498914952638899, −5.39618349317116170933913941120, −4.63232472234054479776499324932, −4.05439596223498545488973813110, −2.86812475331440571900174425096, −1.69259279996982695353933793532, −1.06078946134184179725800490167, 0, 1.06078946134184179725800490167, 1.69259279996982695353933793532, 2.86812475331440571900174425096, 4.05439596223498545488973813110, 4.63232472234054479776499324932, 5.39618349317116170933913941120, 6.48289337539386498914952638899, 7.21063005497497898879488129951, 7.44335916453311745316726202381

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