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Properties

Label 2-115-5.4-c3-0-27
Degree $2$
Conductor $115$
Sign $0.245 + 0.969i$
Analytic cond. $6.78521$
Root an. cond. $2.60484$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.98i·2-s − 9.68i·3-s + 4.04·4-s + (10.8 − 2.74i)5-s + 19.2·6-s − 25.5i·7-s + 23.9i·8-s − 66.8·9-s + (5.45 + 21.5i)10-s − 14.6·11-s − 39.1i·12-s + 70.2i·13-s + 50.8·14-s + (−26.5 − 104. i)15-s − 15.2·16-s − 105. i·17-s + ⋯
L(s)  = 1  + 0.703i·2-s − 1.86i·3-s + 0.505·4-s + (0.969 − 0.245i)5-s + 1.31·6-s − 1.38i·7-s + 1.05i·8-s − 2.47·9-s + (0.172 + 0.681i)10-s − 0.400·11-s − 0.942i·12-s + 1.49i·13-s + 0.970·14-s + (−0.457 − 1.80i)15-s − 0.238·16-s − 1.50i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(6.78521\)
Root analytic conductor: \(2.60484\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3/2),\ 0.245 + 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.58522 - 1.23385i\)
\(L(\frac12)\) \(\approx\) \(1.58522 - 1.23385i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-10.8 + 2.74i)T \)
23 \( 1 - 23iT \)
good2 \( 1 - 1.98iT - 8T^{2} \)
3 \( 1 + 9.68iT - 27T^{2} \)
7 \( 1 + 25.5iT - 343T^{2} \)
11 \( 1 + 14.6T + 1.33e3T^{2} \)
13 \( 1 - 70.2iT - 2.19e3T^{2} \)
17 \( 1 + 105. iT - 4.91e3T^{2} \)
19 \( 1 - 75.6T + 6.85e3T^{2} \)
29 \( 1 + 144.T + 2.43e4T^{2} \)
31 \( 1 - 168.T + 2.97e4T^{2} \)
37 \( 1 + 53.1iT - 5.06e4T^{2} \)
41 \( 1 - 23.8T + 6.89e4T^{2} \)
43 \( 1 - 142. iT - 7.95e4T^{2} \)
47 \( 1 - 190. iT - 1.03e5T^{2} \)
53 \( 1 + 58.7iT - 1.48e5T^{2} \)
59 \( 1 - 342.T + 2.05e5T^{2} \)
61 \( 1 - 459.T + 2.26e5T^{2} \)
67 \( 1 + 282. iT - 3.00e5T^{2} \)
71 \( 1 - 308.T + 3.57e5T^{2} \)
73 \( 1 - 520. iT - 3.89e5T^{2} \)
79 \( 1 - 713.T + 4.93e5T^{2} \)
83 \( 1 - 1.19e3iT - 5.71e5T^{2} \)
89 \( 1 - 35.1T + 7.04e5T^{2} \)
97 \( 1 - 561. iT - 9.12e5T^{2} \)
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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

−13.23696200681225808997394703694, −11.91517306546320342444510619625, −11.12078400385872173829986132296, −9.433452531860973435080858251251, −7.951606242819408712156727794959, −7.07978597333924177359466067489, −6.61194331669900481503984692695, −5.31345064083616356667447537094, −2.45028435304449667741385091188, −1.15828389482635961553218462367, 2.48163262957400714662974225872, 3.42463094557071879461173317674, 5.33565707803368387200772155806, 5.96935844702018918636235823779, 8.388804979925214759995931046491, 9.538192263594409907932048911230, 10.25622672091893410040983988614, 10.85027872735652173203573675809, 11.99451792530826144099402656865, 13.12834356306475289328768915034

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