Properties

Degree $2$
Conductor $114$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s + 9-s + 2·10-s − 4·11-s − 12-s + 2·13-s − 2·15-s + 16-s − 6·17-s + 18-s − 19-s + 2·20-s − 4·22-s − 4·23-s − 24-s − 25-s + 2·26-s − 27-s − 2·29-s − 2·30-s + 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s − 0.852·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.371·29-s − 0.365·30-s + 0.718·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{114} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.387723731\)
\(L(\frac12)\) \(\approx\) \(1.387723731\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
19 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.28366023017921, −17.98867168567639, −17.63600236253260, −16.24119270488123, −15.67252825706852, −14.44025075449834, −13.26471264838449, −12.97088967028241, −11.47795618604631, −10.67649349923182, −9.600878448141020, −7.987570621007900, −6.469250349933732, −5.667556091948056, −4.397900542980190, −2.339734736345750, 2.339734736345750, 4.397900542980190, 5.667556091948056, 6.469250349933732, 7.987570621007900, 9.600878448141020, 10.67649349923182, 11.47795618604631, 12.97088967028241, 13.26471264838449, 14.44025075449834, 15.67252825706852, 16.24119270488123, 17.63600236253260, 17.98867168567639, 19.28366023017921

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