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Properties

Label 2-115-23.22-c6-0-38
Degree $2$
Conductor $115$
Sign $-0.185 + 0.982i$
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  − 0.576·2-s + 45.3·3-s − 63.6·4-s + 55.9i·5-s − 26.1·6-s − 580. i·7-s + 73.6·8-s + 1.32e3·9-s − 32.2i·10-s + 355. i·11-s − 2.88e3·12-s − 3.00e3·13-s + 335. i·14-s + 2.53e3i·15-s + 4.03e3·16-s − 2.91e3i·17-s + ⋯
L(s)  = 1  − 0.0721·2-s + 1.67·3-s − 0.994·4-s + 0.447i·5-s − 0.121·6-s − 1.69i·7-s + 0.143·8-s + 1.81·9-s − 0.0322i·10-s + 0.266i·11-s − 1.67·12-s − 1.36·13-s + 0.122i·14-s + 0.750i·15-s + 0.984·16-s − 0.592i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.185 + 0.982i$
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: \((2,\ 115,\ (\ :3),\ -0.185 + 0.982i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.989665657\)
\(L(\frac12)\) \(\approx\) \(1.989665657\)
\(L(4)\) 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 \( 1 + (1.19e4 + 2.25e3i)T \)
good2 \( 1 + 0.576T + 64T^{2} \)
3 \( 1 - 45.3T + 729T^{2} \)
7 \( 1 + 580. iT - 1.17e5T^{2} \)
11 \( 1 - 355. iT - 1.77e6T^{2} \)
13 \( 1 + 3.00e3T + 4.82e6T^{2} \)
17 \( 1 + 2.91e3iT - 2.41e7T^{2} \)
19 \( 1 + 9.73e3iT - 4.70e7T^{2} \)
29 \( 1 - 2.79e4T + 5.94e8T^{2} \)
31 \( 1 + 2.77e4T + 8.87e8T^{2} \)
37 \( 1 + 7.76e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.05e5T + 4.75e9T^{2} \)
43 \( 1 + 9.72e4iT - 6.32e9T^{2} \)
47 \( 1 - 6.52e4T + 1.07e10T^{2} \)
53 \( 1 - 1.21e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.69e5T + 4.21e10T^{2} \)
61 \( 1 + 2.21e5iT - 5.15e10T^{2} \)
67 \( 1 - 4.81e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.63e5T + 1.28e11T^{2} \)
73 \( 1 + 3.38e5T + 1.51e11T^{2} \)
79 \( 1 + 7.01e5iT - 2.43e11T^{2} \)
83 \( 1 - 2.81e5iT - 3.26e11T^{2} \)
89 \( 1 - 4.02e4iT - 4.96e11T^{2} \)
97 \( 1 - 2.08e5iT - 8.32e11T^{2} \)
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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

−12.57219340130433833891889114465, −10.60260466855532171579907765064, −9.765788061103539799148613782320, −9.010598496720510583463105396877, −7.62054079371726201097002082539, −7.24251631294108136382884574984, −4.62926653849541862430709502176, −3.80500288848834156448990535753, −2.46692250485522661682940498170, −0.54129358602098367656117408167, 1.81937644930588973551388561507, 3.04461060679394692734176241486, 4.41383674017360448252336905474, 5.74638206588698907081857466003, 7.964305210757354186022324979619, 8.390049575235215174077824040653, 9.362928430449071198981414563832, 9.903558932251574423690429222065, 12.24047351078228108223176464891, 12.67446969658199414048944772424

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