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  − 2.29·2-s + 0.676·3-s + 3.28·4-s − 3.14·5-s − 1.55·6-s − 3.61·7-s − 2.95·8-s − 2.54·9-s + 7.22·10-s − 5.00·11-s + 2.22·12-s + 3.46·13-s + 8.30·14-s − 2.12·15-s + 0.221·16-s + 17-s + 5.84·18-s − 2.21·19-s − 10.3·20-s − 2.44·21-s + 11.5·22-s − 4.03·23-s − 1.99·24-s + 4.87·25-s − 7.97·26-s − 3.75·27-s − 11.8·28-s + ⋯
L(s)  = 1  − 1.62·2-s + 0.390·3-s + 1.64·4-s − 1.40·5-s − 0.635·6-s − 1.36·7-s − 1.04·8-s − 0.847·9-s + 2.28·10-s − 1.50·11-s + 0.641·12-s + 0.961·13-s + 2.21·14-s − 0.549·15-s + 0.0554·16-s + 0.242·17-s + 1.37·18-s − 0.508·19-s − 2.30·20-s − 0.533·21-s + 2.45·22-s − 0.840·23-s − 0.408·24-s + 0.974·25-s − 1.56·26-s − 0.721·27-s − 2.24·28-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 \( 1 + 2.29T + 2T^{2} \)
3 \( 1 - 0.676T + 3T^{2} \)
5 \( 1 + 3.14T + 5T^{2} \)
7 \( 1 + 3.61T + 7T^{2} \)
11 \( 1 + 5.00T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
19 \( 1 + 2.21T + 19T^{2} \)
23 \( 1 + 4.03T + 23T^{2} \)
29 \( 1 - 6.02T + 29T^{2} \)
31 \( 1 - 7.26T + 31T^{2} \)
37 \( 1 - 8.52T + 37T^{2} \)
41 \( 1 + 3.54T + 41T^{2} \)
43 \( 1 + 4.72T + 43T^{2} \)
47 \( 1 - 2.95T + 47T^{2} \)
53 \( 1 + 1.45T + 53T^{2} \)
59 \( 1 + 4.01T + 59T^{2} \)
61 \( 1 - 7.07T + 61T^{2} \)
67 \( 1 - 2.33T + 67T^{2} \)
71 \( 1 + 4.74T + 71T^{2} \)
73 \( 1 + 2.40T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + 0.0418T + 83T^{2} \)
89 \( 1 - 4.18T + 89T^{2} \)
97 \( 1 + 0.739T + 97T^{2} \)
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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

−8.038701868798498088673266900566, −7.44499822476617521055152026032, −6.48284461325307860678702931764, −6.04860221295778482504416547919, −4.73808121074598605708359953655, −3.69903830117143919020457014794, −2.99727735089238506723755506390, −2.38020068218420511688157642308, −0.73615488915819050011208999415, 0, 0.73615488915819050011208999415, 2.38020068218420511688157642308, 2.99727735089238506723755506390, 3.69903830117143919020457014794, 4.73808121074598605708359953655, 6.04860221295778482504416547919, 6.48284461325307860678702931764, 7.44499822476617521055152026032, 8.038701868798498088673266900566

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