Properties

Label 2-118-1.1-c1-0-2
Degree $2$
Conductor $118$
Sign $1$
Analytic cond. $0.942234$
Root an. cond. $0.970687$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s − 2·5-s + 2·6-s − 3·7-s + 8-s + 9-s − 2·10-s − 11-s + 2·12-s − 3·13-s − 3·14-s − 4·15-s + 16-s + 7·17-s + 18-s + 4·19-s − 2·20-s − 6·21-s − 22-s + 4·23-s + 2·24-s − 25-s − 3·26-s − 4·27-s − 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.577·12-s − 0.832·13-s − 0.801·14-s − 1.03·15-s + 1/4·16-s + 1.69·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 1.30·21-s − 0.213·22-s + 0.834·23-s + 0.408·24-s − 1/5·25-s − 0.588·26-s − 0.769·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 118 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(118\)    =    \(2 \cdot 59\)
Sign: $1$
Analytic conductor: \(0.942234\)
Root analytic conductor: \(0.970687\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 118,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.684097419\)
\(L(\frac12)\) \(\approx\) \(1.684097419\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
59 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.66307131863104059017579155853, −12.54784789429906755754571018433, −11.87184521188285919790342700281, −10.28222064423934561305183152647, −9.257886156851663271604657828709, −7.898764837731980468726010651930, −7.13529435439983555290616364594, −5.40370004255941961578790717494, −3.64850014003232833399401019862, −2.93278303141535202715010731612, 2.93278303141535202715010731612, 3.64850014003232833399401019862, 5.40370004255941961578790717494, 7.13529435439983555290616364594, 7.898764837731980468726010651930, 9.257886156851663271604657828709, 10.28222064423934561305183152647, 11.87184521188285919790342700281, 12.54784789429906755754571018433, 13.66307131863104059017579155853

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