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  + 0.963·2-s + 3-s − 1.07·4-s − 3.45·5-s + 0.963·6-s + 3.91·7-s − 2.95·8-s + 9-s − 3.32·10-s − 3.42·11-s − 1.07·12-s + 2.91·13-s + 3.77·14-s − 3.45·15-s − 0.710·16-s − 17-s + 0.963·18-s + 7.45·19-s + 3.69·20-s + 3.91·21-s − 3.29·22-s − 8.91·23-s − 2.95·24-s + 6.93·25-s + 2.80·26-s + 27-s − 4.19·28-s + ⋯
L(s)  = 1  + 0.681·2-s + 0.577·3-s − 0.535·4-s − 1.54·5-s + 0.393·6-s + 1.47·7-s − 1.04·8-s + 0.333·9-s − 1.05·10-s − 1.03·11-s − 0.309·12-s + 0.807·13-s + 1.00·14-s − 0.891·15-s − 0.177·16-s − 0.242·17-s + 0.227·18-s + 1.70·19-s + 0.827·20-s + 0.853·21-s − 0.703·22-s − 1.85·23-s − 0.604·24-s + 1.38·25-s + 0.550·26-s + 0.192·27-s − 0.792·28-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 \( 1 - 0.963T + 2T^{2} \)
5 \( 1 + 3.45T + 5T^{2} \)
7 \( 1 - 3.91T + 7T^{2} \)
11 \( 1 + 3.42T + 11T^{2} \)
13 \( 1 - 2.91T + 13T^{2} \)
19 \( 1 - 7.45T + 19T^{2} \)
23 \( 1 + 8.91T + 23T^{2} \)
29 \( 1 + 0.411T + 29T^{2} \)
31 \( 1 + 6.89T + 31T^{2} \)
37 \( 1 + 2.37T + 37T^{2} \)
41 \( 1 - 7.59T + 41T^{2} \)
43 \( 1 + 4.15T + 43T^{2} \)
47 \( 1 - 4.90T + 47T^{2} \)
53 \( 1 - 5.39T + 53T^{2} \)
59 \( 1 + 0.0258T + 59T^{2} \)
61 \( 1 - 5.13T + 61T^{2} \)
67 \( 1 - 0.256T + 67T^{2} \)
71 \( 1 - 5.07T + 71T^{2} \)
73 \( 1 + 4.28T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + 8.39T + 83T^{2} \)
89 \( 1 + 6.35T + 89T^{2} \)
97 \( 1 + 3.92T + 97T^{2} \)
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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

−7.68992696187635526462902857034, −7.09136871309310452872735974061, −5.67229649861635535013736524091, −5.33639664160392080587212692706, −4.41388467553386887326706326140, −4.01035072728825037085052469113, −3.41025014057071759231805855816, −2.48776973228878793277991940129, −1.26429119359557285852917570546, 0, 1.26429119359557285852917570546, 2.48776973228878793277991940129, 3.41025014057071759231805855816, 4.01035072728825037085052469113, 4.41388467553386887326706326140, 5.33639664160392080587212692706, 5.67229649861635535013736524091, 7.09136871309310452872735974061, 7.68992696187635526462902857034

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