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Properties

Label 2-117-1.1-c1-0-4
Degree $2$
Conductor $117$
Sign $-1$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 2·5-s − 4·7-s + 3·8-s + 2·10-s − 4·11-s + 13-s + 4·14-s − 16-s − 2·17-s + 2·20-s + 4·22-s − 25-s − 26-s + 4·28-s + 10·29-s + 4·31-s − 5·32-s + 2·34-s + 8·35-s − 2·37-s − 6·40-s − 6·41-s − 12·43-s + 4·44-s + 9·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.51·7-s + 1.06·8-s + 0.632·10-s − 1.20·11-s + 0.277·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.196·26-s + 0.755·28-s + 1.85·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s + 1.35·35-s − 0.328·37-s − 0.948·40-s − 0.937·41-s − 1.82·43-s + 0.603·44-s + 9/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.08852185820390065936271418820, −12.03518823883711509314697782796, −10.56874363227147435358239986991, −9.880880528421929637795747545010, −8.676495022208841081429592670025, −7.80846844879300530070948409490, −6.52300807259623823318085089821, −4.73928185911620471935520190469, −3.25802804921991097162798742437, 0, 3.25802804921991097162798742437, 4.73928185911620471935520190469, 6.52300807259623823318085089821, 7.80846844879300530070948409490, 8.676495022208841081429592670025, 9.880880528421929637795747545010, 10.56874363227147435358239986991, 12.03518823883711509314697782796, 13.08852185820390065936271418820

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