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  − 2.45·2-s + 3-s + 4.04·4-s + 2.82·5-s − 2.45·6-s + 4.27·7-s − 5.03·8-s + 9-s − 6.94·10-s − 3.19·11-s + 4.04·12-s + 0.537·13-s − 10.5·14-s + 2.82·15-s + 4.28·16-s − 17-s − 2.45·18-s − 4.65·19-s + 11.4·20-s + 4.27·21-s + 7.86·22-s − 1.04·23-s − 5.03·24-s + 2.97·25-s − 1.32·26-s + 27-s + 17.3·28-s + ⋯
L(s)  = 1  − 1.73·2-s + 0.577·3-s + 2.02·4-s + 1.26·5-s − 1.00·6-s + 1.61·7-s − 1.78·8-s + 0.333·9-s − 2.19·10-s − 0.963·11-s + 1.16·12-s + 0.148·13-s − 2.81·14-s + 0.729·15-s + 1.07·16-s − 0.242·17-s − 0.579·18-s − 1.06·19-s + 2.55·20-s + 0.933·21-s + 1.67·22-s − 0.217·23-s − 1.02·24-s + 0.594·25-s − 0.259·26-s + 0.192·27-s + 3.27·28-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 \( 1 + 2.45T + 2T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 - 4.27T + 7T^{2} \)
11 \( 1 + 3.19T + 11T^{2} \)
13 \( 1 - 0.537T + 13T^{2} \)
19 \( 1 + 4.65T + 19T^{2} \)
23 \( 1 + 1.04T + 23T^{2} \)
29 \( 1 + 9.34T + 29T^{2} \)
31 \( 1 + 8.67T + 31T^{2} \)
37 \( 1 + 5.77T + 37T^{2} \)
41 \( 1 + 7.18T + 41T^{2} \)
43 \( 1 + 1.84T + 43T^{2} \)
47 \( 1 + 9.23T + 47T^{2} \)
53 \( 1 - 1.01T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 2.60T + 61T^{2} \)
67 \( 1 - 2.87T + 67T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 - 9.34T + 73T^{2} \)
79 \( 1 - 3.82T + 79T^{2} \)
83 \( 1 + 6.64T + 83T^{2} \)
89 \( 1 + 8.76T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−7.901402236656570270287737402446, −7.11188653329397362067661546209, −6.38607467156007044858990128364, −5.44814146157978437393411288754, −4.93212497802848803393873773857, −3.67403105048927443401771646082, −2.41573440712306452409157987841, −1.83969349986570414606696281576, −1.62349738882136216876935302500, 0, 1.62349738882136216876935302500, 1.83969349986570414606696281576, 2.41573440712306452409157987841, 3.67403105048927443401771646082, 4.93212497802848803393873773857, 5.44814146157978437393411288754, 6.38607467156007044858990128364, 7.11188653329397362067661546209, 7.901402236656570270287737402446

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