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Properties

Label 2-119-119.118-c0-0-2
Degree $2$
Conductor $119$
Sign $1$
Analytic cond. $0.0593887$
Root an. cond. $0.243698$
Motivic weight $0$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $1$
Analytic conductor: \(0.0593887\)
Root analytic conductor: \(0.243698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (118, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 119,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6302811707\)
\(L(\frac12)\) \(\approx\) \(0.6302811707\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.60067471336295545512712433556, −13.00791308066877707094697030464, −11.97448861661235573494853187038, −10.52527424062377841113145891281, −9.578200214180807123496499712112, −8.770078937487255232414253344988, −6.42543460903241917673966198786, −5.93263218695135416564774589031, −4.77696768335113156866800341807, −2.86242758482639499100631131705, 2.86242758482639499100631131705, 4.77696768335113156866800341807, 5.93263218695135416564774589031, 6.42543460903241917673966198786, 8.770078937487255232414253344988, 9.578200214180807123496499712112, 10.52527424062377841113145891281, 11.97448861661235573494853187038, 13.00791308066877707094697030464, 13.60067471336295545512712433556

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