Properties

Degree 2
Conductor $ 7 \cdot 17 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s − 1.61·3-s + 1.61·4-s + 0.618·5-s + 2.61·6-s + 7-s − 8-s + 1.61·9-s − 1.00·10-s − 2.61·12-s − 1.61·14-s − 1.00·15-s + 17-s − 2.61·18-s + 1.00·20-s − 1.61·21-s + 1.61·24-s − 0.618·25-s − 27-s + 1.61·28-s + 1.61·30-s + 0.618·31-s + 32-s − 1.61·34-s + 0.618·35-s + 2.61·36-s − 0.618·40-s + ⋯
L(s)  = 1  − 1.61·2-s − 1.61·3-s + 1.61·4-s + 0.618·5-s + 2.61·6-s + 7-s − 8-s + 1.61·9-s − 1.00·10-s − 2.61·12-s − 1.61·14-s − 1.00·15-s + 17-s − 2.61·18-s + 1.00·20-s − 1.61·21-s + 1.61·24-s − 0.618·25-s − 27-s + 1.61·28-s + 1.61·30-s + 0.618·31-s + 32-s − 1.61·34-s + 0.618·35-s + 2.61·36-s − 0.618·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(119\)    =    \(7 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{119} (118, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 119,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.2343847461\)
\(L(\frac12)\)  \(\approx\)  \(0.2343847461\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;17\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 - T \)
17 \( 1 - T \)
good2 \( 1 + 1.61T + T^{2} \)
3 \( 1 + 1.61T + T^{2} \)
5 \( 1 - 0.618T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.61T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 0.618T + T^{2} \)
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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

−13.65194338558675342126772884310, −12.04645147468382634720510609770, −11.44893583765498948948023432146, −10.44337523099519291188430246284, −9.878008452330803346119504481607, −8.439959401615032609354256443612, −7.30241031392590951611595560944, −6.12094459230689769919386566111, −4.97597257629778139191185706419, −1.52402899467920246594218058938, 1.52402899467920246594218058938, 4.97597257629778139191185706419, 6.12094459230689769919386566111, 7.30241031392590951611595560944, 8.439959401615032609354256443612, 9.878008452330803346119504481607, 10.44337523099519291188430246284, 11.44893583765498948948023432146, 12.04645147468382634720510609770, 13.65194338558675342126772884310

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