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  − 1.58·2-s + 3-s + 0.499·4-s − 3.53·5-s − 1.58·6-s − 3.85·7-s + 2.37·8-s + 9-s + 5.59·10-s − 4.80·11-s + 0.499·12-s + 2.42·13-s + 6.08·14-s − 3.53·15-s − 4.74·16-s − 17-s − 1.58·18-s + 1.75·19-s − 1.76·20-s − 3.85·21-s + 7.58·22-s − 7.27·23-s + 2.37·24-s + 7.51·25-s − 3.83·26-s + 27-s − 1.92·28-s + ⋯
L(s)  = 1  − 1.11·2-s + 0.577·3-s + 0.249·4-s − 1.58·5-s − 0.645·6-s − 1.45·7-s + 0.838·8-s + 0.333·9-s + 1.76·10-s − 1.44·11-s + 0.144·12-s + 0.672·13-s + 1.62·14-s − 0.913·15-s − 1.18·16-s − 0.242·17-s − 0.372·18-s + 0.402·19-s − 0.394·20-s − 0.840·21-s + 1.61·22-s − 1.51·23-s + 0.484·24-s + 1.50·25-s − 0.751·26-s + 0.192·27-s − 0.363·28-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 \( 1 + 1.58T + 2T^{2} \)
5 \( 1 + 3.53T + 5T^{2} \)
7 \( 1 + 3.85T + 7T^{2} \)
11 \( 1 + 4.80T + 11T^{2} \)
13 \( 1 - 2.42T + 13T^{2} \)
19 \( 1 - 1.75T + 19T^{2} \)
23 \( 1 + 7.27T + 23T^{2} \)
29 \( 1 + 1.01T + 29T^{2} \)
31 \( 1 + 0.736T + 31T^{2} \)
37 \( 1 - 0.762T + 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 7.99T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 + 7.96T + 53T^{2} \)
59 \( 1 - 0.229T + 59T^{2} \)
61 \( 1 - 7.63T + 61T^{2} \)
67 \( 1 - 2.68T + 67T^{2} \)
71 \( 1 - 5.96T + 71T^{2} \)
73 \( 1 + 5.15T + 73T^{2} \)
79 \( 1 - 3.52T + 79T^{2} \)
83 \( 1 + 3.41T + 83T^{2} \)
89 \( 1 - 5.27T + 89T^{2} \)
97 \( 1 + 9.65T + 97T^{2} \)
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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

−7.66438754494984142440560645356, −7.27724213066540493561426507941, −6.38430755946825155877947794478, −5.45266741309467217042914993813, −4.29406928527893302161010256231, −3.87902168307186986786973376440, −3.07351627907605135588722953305, −2.26425232156516421140407588469, −0.75505805864873894824777186711, 0, 0.75505805864873894824777186711, 2.26425232156516421140407588469, 3.07351627907605135588722953305, 3.87902168307186986786973376440, 4.29406928527893302161010256231, 5.45266741309467217042914993813, 6.38430755946825155877947794478, 7.27724213066540493561426507941, 7.66438754494984142440560645356

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