Properties

Degree 2
Conductor $ 17 \cdot 353 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.31·2-s − 0.975·3-s + 3.35·4-s − 3.23·5-s + 2.25·6-s + 5.09·7-s − 3.14·8-s − 2.04·9-s + 7.47·10-s + 3.94·11-s − 3.27·12-s + 1.47·13-s − 11.7·14-s + 3.15·15-s + 0.565·16-s + 17-s + 4.74·18-s + 0.961·19-s − 10.8·20-s − 4.96·21-s − 9.14·22-s + 0.0822·23-s + 3.06·24-s + 5.43·25-s − 3.41·26-s + 4.92·27-s + 17.1·28-s + ⋯
L(s)  = 1  − 1.63·2-s − 0.562·3-s + 1.67·4-s − 1.44·5-s + 0.921·6-s + 1.92·7-s − 1.11·8-s − 0.683·9-s + 2.36·10-s + 1.19·11-s − 0.945·12-s + 0.409·13-s − 3.15·14-s + 0.813·15-s + 0.141·16-s + 0.242·17-s + 1.11·18-s + 0.220·19-s − 2.42·20-s − 1.08·21-s − 1.94·22-s + 0.0171·23-s + 0.626·24-s + 1.08·25-s − 0.670·26-s + 0.947·27-s + 3.23·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6001\)    =    \(17 \cdot 353\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6001} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 6001,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;353\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{17,\;353\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 \( 1 - T \)
353 \( 1 - T \)
good2 \( 1 + 2.31T + 2T^{2} \)
3 \( 1 + 0.975T + 3T^{2} \)
5 \( 1 + 3.23T + 5T^{2} \)
7 \( 1 - 5.09T + 7T^{2} \)
11 \( 1 - 3.94T + 11T^{2} \)
13 \( 1 - 1.47T + 13T^{2} \)
19 \( 1 - 0.961T + 19T^{2} \)
23 \( 1 - 0.0822T + 23T^{2} \)
29 \( 1 - 1.83T + 29T^{2} \)
31 \( 1 + 1.55T + 31T^{2} \)
37 \( 1 + 2.94T + 37T^{2} \)
41 \( 1 + 5.60T + 41T^{2} \)
43 \( 1 + 4.26T + 43T^{2} \)
47 \( 1 + 2.32T + 47T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 + 3.51T + 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + 7.16T + 67T^{2} \)
71 \( 1 + 3.49T + 71T^{2} \)
73 \( 1 - 3.92T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + 9.02T + 89T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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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

−8.039305517944810514676414422293, −7.25259583470462881608630561564, −6.73263483497439378092709104162, −5.68722756806493738642400043360, −4.78686059472751328954382179600, −4.13853354394603018319134516392, −3.08504559671358592538840449240, −1.70010184677534548117803447308, −1.11586482626326427059863755255, 0, 1.11586482626326427059863755255, 1.70010184677534548117803447308, 3.08504559671358592538840449240, 4.13853354394603018319134516392, 4.78686059472751328954382179600, 5.68722756806493738642400043360, 6.73263483497439378092709104162, 7.25259583470462881608630561564, 8.039305517944810514676414422293

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