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Properties

Label 2-115-5.3-c2-0-19
Degree $2$
Conductor $115$
Sign $-0.660 + 0.750i$
Analytic cond. $3.13352$
Root an. cond. $1.77017$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.63 − 2.63i)2-s + (−0.614 − 0.614i)3-s − 9.92i·4-s + (−4.07 + 2.89i)5-s − 3.24·6-s + (1.62 − 1.62i)7-s + (−15.6 − 15.6i)8-s − 8.24i·9-s + (−3.12 + 18.3i)10-s + 12.6·11-s + (−6.09 + 6.09i)12-s + (13.1 + 13.1i)13-s − 8.57i·14-s + (4.28 + 0.728i)15-s − 42.7·16-s + (2.73 − 2.73i)17-s + ⋯
L(s)  = 1  + (1.31 − 1.31i)2-s + (−0.204 − 0.204i)3-s − 2.48i·4-s + (−0.815 + 0.578i)5-s − 0.540·6-s + (0.232 − 0.232i)7-s + (−1.95 − 1.95i)8-s − 0.916i·9-s + (−0.312 + 1.83i)10-s + 1.15·11-s + (−0.507 + 0.507i)12-s + (1.01 + 1.01i)13-s − 0.612i·14-s + (0.285 + 0.0485i)15-s − 2.67·16-s + (0.161 − 0.161i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.660 + 0.750i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ -0.660 + 0.750i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.935630 - 2.07072i\)
\(L(\frac12)\) \(\approx\) \(0.935630 - 2.07072i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.07 - 2.89i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good2 \( 1 + (-2.63 + 2.63i)T - 4iT^{2} \)
3 \( 1 + (0.614 + 0.614i)T + 9iT^{2} \)
7 \( 1 + (-1.62 + 1.62i)T - 49iT^{2} \)
11 \( 1 - 12.6T + 121T^{2} \)
13 \( 1 + (-13.1 - 13.1i)T + 169iT^{2} \)
17 \( 1 + (-2.73 + 2.73i)T - 289iT^{2} \)
19 \( 1 - 26.9iT - 361T^{2} \)
29 \( 1 - 4.50iT - 841T^{2} \)
31 \( 1 + 38.0T + 961T^{2} \)
37 \( 1 + (-16.1 + 16.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 73.7T + 1.68e3T^{2} \)
43 \( 1 + (8.28 + 8.28i)T + 1.84e3iT^{2} \)
47 \( 1 + (-30.7 + 30.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-52.1 - 52.1i)T + 2.80e3iT^{2} \)
59 \( 1 + 34.1iT - 3.48e3T^{2} \)
61 \( 1 + 2.91T + 3.72e3T^{2} \)
67 \( 1 + (55.9 - 55.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 3.68T + 5.04e3T^{2} \)
73 \( 1 + (-86.8 - 86.8i)T + 5.32e3iT^{2} \)
79 \( 1 - 52.8iT - 6.24e3T^{2} \)
83 \( 1 + (101. + 101. i)T + 6.88e3iT^{2} \)
89 \( 1 + 98.9iT - 7.92e3T^{2} \)
97 \( 1 + (-1.90 + 1.90i)T - 9.40e3iT^{2} \)
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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

−12.66303600886717431431585855213, −11.77312393809989755654058212830, −11.43829596249417405644979377808, −10.31060122769297766312161350946, −9.002638633213285170231600480356, −6.91252271543637810130176426748, −5.92003365444513692086572405246, −4.06332363882323828098151315994, −3.57666603032619257820527654543, −1.42790639382048308223840975176, 3.53127612234937543705044078907, 4.66929713534406634514530868483, 5.56248833969923676020511241597, 6.90134614818380212752433795715, 8.030562343415461348409284948130, 8.810736922915282795567463667114, 11.08031133862296560197899941051, 11.94100008523965940755898240508, 13.03388583710523791352985044664, 13.66622142043064869903338786669

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