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Properties

Label 2-115-115.114-c4-0-5
Degree $2$
Conductor $115$
Sign $-0.990 + 0.134i$
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  + 6.07i·2-s − 5.74i·3-s − 20.9·4-s + (19.8 + 15.2i)5-s + 34.9·6-s − 44.4·7-s − 29.8i·8-s + 47.9·9-s + (−92.6 + 120. i)10-s + 28.8i·11-s + 120. i·12-s + 250. i·13-s − 269. i·14-s + (87.6 − 113. i)15-s − 153.·16-s − 450.·17-s + ⋯
L(s)  = 1  + 1.51i·2-s − 0.638i·3-s − 1.30·4-s + (0.792 + 0.609i)5-s + 0.970·6-s − 0.906·7-s − 0.465i·8-s + 0.592·9-s + (−0.926 + 1.20i)10-s + 0.238i·11-s + 0.834i·12-s + 1.48i·13-s − 1.37i·14-s + (0.389 − 0.506i)15-s − 0.599·16-s − 1.55·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.990 + 0.134i$
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: \((2,\ 115,\ (\ :2),\ -0.990 + 0.134i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0900503 - 1.33222i\)
\(L(\frac12)\) \(\approx\) \(0.0900503 - 1.33222i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-19.8 - 15.2i)T \)
23 \( 1 + (371. - 376. i)T \)
good2 \( 1 - 6.07iT - 16T^{2} \)
3 \( 1 + 5.74iT - 81T^{2} \)
7 \( 1 + 44.4T + 2.40e3T^{2} \)
11 \( 1 - 28.8iT - 1.46e4T^{2} \)
13 \( 1 - 250. iT - 2.85e4T^{2} \)
17 \( 1 + 450.T + 8.35e4T^{2} \)
19 \( 1 + 109. iT - 1.30e5T^{2} \)
29 \( 1 + 103.T + 7.07e5T^{2} \)
31 \( 1 - 874.T + 9.23e5T^{2} \)
37 \( 1 + 494.T + 1.87e6T^{2} \)
41 \( 1 - 563.T + 2.82e6T^{2} \)
43 \( 1 - 1.72e3T + 3.41e6T^{2} \)
47 \( 1 - 3.28e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.26e3T + 7.89e6T^{2} \)
59 \( 1 - 4.67e3T + 1.21e7T^{2} \)
61 \( 1 - 6.20e3iT - 1.38e7T^{2} \)
67 \( 1 - 6.04e3T + 2.01e7T^{2} \)
71 \( 1 - 2.62e3T + 2.54e7T^{2} \)
73 \( 1 + 1.55e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.54e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.05e3T + 4.74e7T^{2} \)
89 \( 1 + 1.18e4iT - 6.27e7T^{2} \)
97 \( 1 - 8.50e3T + 8.85e7T^{2} \)
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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

−13.64578760011755261850201018966, −12.90080597633787117615917114595, −11.36152654838356503129874362738, −9.796637997145844505398386491361, −8.993939254463844635689705371937, −7.42237137580842385717786705275, −6.65194869481599019865674129282, −6.18478437432020025029897192591, −4.44177840194190371230429409842, −2.13907854265670022084796164582, 0.56170851346014634904512432419, 2.28363952782620619550988933942, 3.65854045441226126845432523157, 4.88113228655595901365720054503, 6.42979661828591897663180102641, 8.567215904987235142566191167746, 9.672110966201181969956011691587, 10.14439401655419822618571245816, 11.01736908359041916640290669916, 12.59030671349804963371198095925

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