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Properties

Label 2-115-23.22-c6-0-24
Degree $2$
Conductor $115$
Sign $0.997 - 0.0716i$
Analytic cond. $26.4562$
Root an. cond. $5.14356$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.12·2-s − 10.9·3-s − 37.7·4-s + 55.9i·5-s − 56.2·6-s − 205. i·7-s − 521.·8-s − 608.·9-s + 286. i·10-s − 349. i·11-s + 415.·12-s + 4.18e3·13-s − 1.05e3i·14-s − 614. i·15-s − 250.·16-s + 4.98e3i·17-s + ⋯
L(s)  = 1  + 0.640·2-s − 0.406·3-s − 0.590·4-s + 0.447i·5-s − 0.260·6-s − 0.600i·7-s − 1.01·8-s − 0.834·9-s + 0.286i·10-s − 0.262i·11-s + 0.240·12-s + 1.90·13-s − 0.384i·14-s − 0.181i·15-s − 0.0611·16-s + 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0716i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.997 - 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.997 - 0.0716i$
Analytic conductor: \(26.4562\)
Root analytic conductor: \(5.14356\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3),\ 0.997 - 0.0716i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.728869904\)
\(L(\frac12)\) \(\approx\) \(1.728869904\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 55.9iT \)
23 \( 1 + (-871. - 1.21e4i)T \)
good2 \( 1 - 5.12T + 64T^{2} \)
3 \( 1 + 10.9T + 729T^{2} \)
7 \( 1 + 205. iT - 1.17e5T^{2} \)
11 \( 1 + 349. iT - 1.77e6T^{2} \)
13 \( 1 - 4.18e3T + 4.82e6T^{2} \)
17 \( 1 - 4.98e3iT - 2.41e7T^{2} \)
19 \( 1 + 5.70e3iT - 4.70e7T^{2} \)
29 \( 1 - 4.16e4T + 5.94e8T^{2} \)
31 \( 1 - 1.80e3T + 8.87e8T^{2} \)
37 \( 1 + 8.94e4iT - 2.56e9T^{2} \)
41 \( 1 - 3.50e4T + 4.75e9T^{2} \)
43 \( 1 - 1.32e5iT - 6.32e9T^{2} \)
47 \( 1 - 1.01e5T + 1.07e10T^{2} \)
53 \( 1 + 2.33e4iT - 2.21e10T^{2} \)
59 \( 1 + 1.88e5T + 4.21e10T^{2} \)
61 \( 1 + 3.16e5iT - 5.15e10T^{2} \)
67 \( 1 - 5.73e4iT - 9.04e10T^{2} \)
71 \( 1 + 1.51e5T + 1.28e11T^{2} \)
73 \( 1 - 7.36e3T + 1.51e11T^{2} \)
79 \( 1 + 4.67e5iT - 2.43e11T^{2} \)
83 \( 1 - 7.48e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.13e6iT - 4.96e11T^{2} \)
97 \( 1 + 1.40e6iT - 8.32e11T^{2} \)
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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

−12.58808040591146664711815479536, −11.29650313879375694859799058653, −10.68171614655078663019277238075, −9.124126920902649672504857232475, −8.172281975429938329964616371994, −6.41366324535654701656010714346, −5.67545883463949924599565246203, −4.18184679404871848097152717605, −3.16554052517075293847881977537, −0.811361329186093920457018109447, 0.795598533272140212966280504956, 2.97348248401223595225339999144, 4.39834207225001412165887680619, 5.52608785965771866719495844778, 6.30485274299032671406231135567, 8.442853040335247229140620509279, 8.892156421112849372713320156187, 10.39834398791645207976705766442, 11.76797103745183218050735606663, 12.30459613276320111728839878129

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