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Properties

Label 2-115-115.114-c4-0-45
Degree $2$
Conductor $115$
Sign $0.833 - 0.552i$
Analytic cond. $11.8875$
Root an. cond. $3.44783$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.76i·2-s − 12.7i·3-s − 29.8·4-s + (−13.2 + 21.2i)5-s − 86.4·6-s + 5.97·7-s + 93.4i·8-s − 82.1·9-s + (143. + 89.3i)10-s + 18.7i·11-s + 380. i·12-s + 30.6i·13-s − 40.4i·14-s + (271. + 168. i)15-s + 155.·16-s − 280.·17-s + ⋯
L(s)  = 1  − 1.69i·2-s − 1.41i·3-s − 1.86·4-s + (−0.528 + 0.849i)5-s − 2.40·6-s + 0.121·7-s + 1.46i·8-s − 1.01·9-s + (1.43 + 0.893i)10-s + 0.155i·11-s + 2.64i·12-s + 0.181i·13-s − 0.206i·14-s + (1.20 + 0.749i)15-s + 0.608·16-s − 0.971·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.833 - 0.552i$
Analytic conductor: \(11.8875\)
Root analytic conductor: \(3.44783\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :2),\ 0.833 - 0.552i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.265094 + 0.0799513i\)
\(L(\frac12)\) \(\approx\) \(0.265094 + 0.0799513i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (13.2 - 21.2i)T \)
23 \( 1 + (481. + 219. i)T \)
good2 \( 1 + 6.76iT - 16T^{2} \)
3 \( 1 + 12.7iT - 81T^{2} \)
7 \( 1 - 5.97T + 2.40e3T^{2} \)
11 \( 1 - 18.7iT - 1.46e4T^{2} \)
13 \( 1 - 30.6iT - 2.85e4T^{2} \)
17 \( 1 + 280.T + 8.35e4T^{2} \)
19 \( 1 - 93.6iT - 1.30e5T^{2} \)
29 \( 1 - 378.T + 7.07e5T^{2} \)
31 \( 1 - 25.0T + 9.23e5T^{2} \)
37 \( 1 + 2.50e3T + 1.87e6T^{2} \)
41 \( 1 - 182.T + 2.82e6T^{2} \)
43 \( 1 - 3.03e3T + 3.41e6T^{2} \)
47 \( 1 - 2.99e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.12e3T + 7.89e6T^{2} \)
59 \( 1 + 495.T + 1.21e7T^{2} \)
61 \( 1 + 3.23e3iT - 1.38e7T^{2} \)
67 \( 1 + 5.93e3T + 2.01e7T^{2} \)
71 \( 1 + 8.19e3T + 2.54e7T^{2} \)
73 \( 1 + 6.90e3iT - 2.83e7T^{2} \)
79 \( 1 - 3.46e3iT - 3.89e7T^{2} \)
83 \( 1 + 7.71e3T + 4.74e7T^{2} \)
89 \( 1 - 2.86e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.52e3T + 8.85e7T^{2} \)
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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

−11.99597261157795625184799525260, −11.12193588216989091150538645877, −10.22776108075313501481106838785, −8.745663110581633329979635782230, −7.54168486046294545046189181485, −6.40642221657024911400631345893, −4.23149598696874856633382391412, −2.76015967441899488910601670980, −1.71974412646495322626749410634, −0.11974346179305530082886418936, 3.96095036413996811455380213674, 4.78907557384286616327298581306, 5.72756899710910252093142014399, 7.28324618371942487806551072564, 8.550604601254261833946188462114, 9.056648681887890549032867333872, 10.30527243236774313570614157502, 11.67499685370870310821050072302, 13.17569682843161815015939783839, 14.21320620194939977407638807099

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