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  + 0.618·2-s + 0.618·3-s − 0.618·4-s − 1.61·5-s + 0.381·6-s + 7-s − 8-s − 0.618·9-s − 1.00·10-s − 0.381·12-s + 0.618·14-s − 1.00·15-s + 17-s − 0.381·18-s + 0.999·20-s + 0.618·21-s − 0.618·24-s + 1.61·25-s − 27-s − 0.618·28-s − 0.618·30-s − 1.61·31-s + 0.999·32-s + 0.618·34-s − 1.61·35-s + 0.381·36-s + 1.61·40-s + ⋯
L(s)  = 1  + 0.618·2-s + 0.618·3-s − 0.618·4-s − 1.61·5-s + 0.381·6-s + 7-s − 8-s − 0.618·9-s − 1.00·10-s − 0.381·12-s + 0.618·14-s − 1.00·15-s + 17-s − 0.381·18-s + 0.999·20-s + 0.618·21-s − 0.618·24-s + 1.61·25-s − 27-s − 0.618·28-s − 0.618·30-s − 1.61·31-s + 0.999·32-s + 0.618·34-s − 1.61·35-s + 0.381·36-s + 1.61·40-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  =  \((2,\ 119,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.6840384900\)
\(L(\frac12)\)  \(\approx\)  \(0.6840384900\)
\(L(1)\)   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 \( 1 - 0.618T + T^{2} \)
3 \( 1 - 0.618T + T^{2} \)
5 \( 1 + 1.61T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.61T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.61T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.618T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.61T + T^{2} \)
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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

−14.04573251757204190644351240073, −12.68729914443010867183501235312, −11.84702716241586137927181515730, −10.99785240360884651423191253530, −9.197151922943741932582102142604, −8.252672509741494813962148682655, −7.59622314204297360302731343397, −5.48989736619258142965789236319, −4.26497223589162521783933658687, −3.27448017805867294444555410081, 3.27448017805867294444555410081, 4.26497223589162521783933658687, 5.48989736619258142965789236319, 7.59622314204297360302731343397, 8.252672509741494813962148682655, 9.197151922943741932582102142604, 10.99785240360884651423191253530, 11.84702716241586137927181515730, 12.68729914443010867183501235312, 14.04573251757204190644351240073

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