Properties

Label 2-115-23.18-c1-0-1
Degree $2$
Conductor $115$
Sign $-0.143 - 0.989i$
Analytic cond. $0.918279$
Root an. cond. $0.958269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.29 + 0.380i)2-s + (−1.87 + 2.16i)3-s + (−0.145 − 0.0937i)4-s + (−0.142 + 0.989i)5-s + (−3.25 + 2.09i)6-s + (1.66 + 3.64i)7-s + (−1.92 − 2.21i)8-s + (−0.740 − 5.15i)9-s + (−0.561 + 1.22i)10-s + (5.32 − 1.56i)11-s + (0.476 − 0.139i)12-s + (0.162 − 0.356i)13-s + (0.771 + 5.36i)14-s + (−1.87 − 2.16i)15-s + (−1.50 − 3.29i)16-s + (−0.659 + 0.423i)17-s + ⋯
L(s)  = 1  + (0.916 + 0.269i)2-s + (−1.08 + 1.24i)3-s + (−0.0729 − 0.0468i)4-s + (−0.0636 + 0.442i)5-s + (−1.32 + 0.854i)6-s + (0.629 + 1.37i)7-s + (−0.680 − 0.784i)8-s + (−0.246 − 1.71i)9-s + (−0.177 + 0.388i)10-s + (1.60 − 0.471i)11-s + (0.137 − 0.0403i)12-s + (0.0451 − 0.0987i)13-s + (0.206 + 1.43i)14-s + (−0.484 − 0.558i)15-s + (−0.376 − 0.823i)16-s + (−0.159 + 0.102i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.143 - 0.989i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ -0.143 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.742926 + 0.858836i\)
\(L(\frac12)\) \(\approx\) \(0.742926 + 0.858836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (-4.73 - 0.783i)T \)
good2 \( 1 + (-1.29 - 0.380i)T + (1.68 + 1.08i)T^{2} \)
3 \( 1 + (1.87 - 2.16i)T + (-0.426 - 2.96i)T^{2} \)
7 \( 1 + (-1.66 - 3.64i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-5.32 + 1.56i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-0.162 + 0.356i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.659 - 0.423i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (0.661 + 0.425i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (0.130 - 0.0840i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (2.02 + 2.33i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (1.58 + 11.0i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (1.44 - 10.0i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (1.99 - 2.30i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 - 2.46T + 47T^{2} \)
53 \( 1 + (2.03 + 4.44i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-4.65 + 10.1i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (4.03 + 4.65i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (5.93 + 1.74i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (-4.07 - 1.19i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (0.795 + 0.511i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (0.726 - 1.59i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-1.38 - 9.66i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-5.26 + 6.07i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-1.60 + 11.1i)T + (-93.0 - 27.3i)T^{2} \)
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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

−14.28999128629930698226986529465, −12.65822891924038527271888277569, −11.63860525467620918339082520805, −11.12135258419788296464268277340, −9.602962061165700151556167176861, −8.895096401970780793858384305919, −6.47436954094112311860405956373, −5.70011820969388663597845854762, −4.76352794637077942523390041845, −3.60138457119017634039531737455, 1.34902075979349022788815147091, 4.07051842438134741079234491632, 5.08030842561796034607063950107, 6.51454660089060569251612105258, 7.40303161100663783220346188635, 8.827076332216665504957544667590, 10.71130497760929917326412276184, 11.74494054263670280875724378622, 12.18863787905886878287707562151, 13.27905329474397858076693914572

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