Properties

Label 2-115-115.114-c2-0-10
Degree $2$
Conductor $115$
Sign $1$
Analytic cond. $3.13352$
Root an. cond. $1.77017$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s − 5·5-s + 9·7-s + 9·9-s + 16·16-s − 11·17-s − 20·20-s + 23·23-s + 25·25-s + 36·28-s − 57·29-s − 53·31-s − 45·35-s + 36·36-s − 51·37-s − 33·41-s + 6·43-s − 45·45-s + 32·49-s + 101·53-s + 3·59-s + 81·63-s + 64·64-s − 111·67-s − 44·68-s + 27·71-s − 80·80-s + ⋯
L(s)  = 1  + 4-s − 5-s + 9/7·7-s + 9-s + 16-s − 0.647·17-s − 20-s + 23-s + 25-s + 9/7·28-s − 1.96·29-s − 1.70·31-s − 9/7·35-s + 36-s − 1.37·37-s − 0.804·41-s + 6/43·43-s − 45-s + 0.653·49-s + 1.90·53-s + 3/59·59-s + 9/7·63-s + 64-s − 1.65·67-s − 0.647·68-s + 0.380·71-s − 80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $1$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{115} (114, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.660123602\)
\(L(\frac12)\) \(\approx\) \(1.660123602\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + p T \)
23 \( 1 - p T \)
good2 \( ( 1 - p T )( 1 + p T ) \)
3 \( ( 1 - p T )( 1 + p T ) \)
7 \( 1 - 9 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( ( 1 - p T )( 1 + p T ) \)
17 \( 1 + 11 T + p^{2} T^{2} \)
19 \( ( 1 - p T )( 1 + p T ) \)
29 \( 1 + 57 T + p^{2} T^{2} \)
31 \( 1 + 53 T + p^{2} T^{2} \)
37 \( 1 + 51 T + p^{2} T^{2} \)
41 \( 1 + 33 T + p^{2} T^{2} \)
43 \( 1 - 6 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( 1 - 101 T + p^{2} T^{2} \)
59 \( 1 - 3 T + p^{2} T^{2} \)
61 \( ( 1 - p T )( 1 + p T ) \)
67 \( 1 + 111 T + p^{2} T^{2} \)
71 \( 1 - 27 T + p^{2} T^{2} \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( ( 1 - p T )( 1 + p T ) \)
83 \( 1 - 41 T + p^{2} T^{2} \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 - 174 T + p^{2} T^{2} \)
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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

−13.14537281575095023826047527718, −12.08378315987915710246843163531, −11.21399299337332141202581124704, −10.62452007771714500090055854360, −8.857917611195818748694982637093, −7.57278308374617803582187074657, −7.05968412688741032916290959711, −5.18167981241423476593399795811, −3.78322829743545174166204612708, −1.75247084218293106929757736097, 1.75247084218293106929757736097, 3.78322829743545174166204612708, 5.18167981241423476593399795811, 7.05968412688741032916290959711, 7.57278308374617803582187074657, 8.857917611195818748694982637093, 10.62452007771714500090055854360, 11.21399299337332141202581124704, 12.08378315987915710246843163531, 13.14537281575095023826047527718

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