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Properties

Label 2-115-5.4-c1-0-6
Degree $2$
Conductor $115$
Sign $0.995 - 0.0938i$
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  + 0.751i·2-s − 0.580i·3-s + 1.43·4-s + (−0.209 − 2.22i)5-s + 0.435·6-s − 0.315i·7-s + 2.58i·8-s + 2.66·9-s + (1.67 − 0.157i)10-s − 4.34·11-s − 0.833i·12-s + 4.58i·13-s + 0.236·14-s + (−1.29 + 0.121i)15-s + 0.933·16-s − 0.917i·17-s + ⋯
L(s)  = 1  + 0.531i·2-s − 0.335i·3-s + 0.717·4-s + (−0.0938 − 0.995i)5-s + 0.177·6-s − 0.119i·7-s + 0.912i·8-s + 0.887·9-s + (0.528 − 0.0498i)10-s − 1.31·11-s − 0.240i·12-s + 1.27i·13-s + 0.0632·14-s + (−0.333 + 0.0314i)15-s + 0.233·16-s − 0.222i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.995 - 0.0938i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ 0.995 - 0.0938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18782 + 0.0558350i\)
\(L(\frac12)\) \(\approx\) \(1.18782 + 0.0558350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.209 + 2.22i)T \)
23 \( 1 + iT \)
good2 \( 1 - 0.751iT - 2T^{2} \)
3 \( 1 + 0.580iT - 3T^{2} \)
7 \( 1 + 0.315iT - 7T^{2} \)
11 \( 1 + 4.34T + 11T^{2} \)
13 \( 1 - 4.58iT - 13T^{2} \)
17 \( 1 + 0.917iT - 17T^{2} \)
19 \( 1 + 2.76T + 19T^{2} \)
29 \( 1 + 7.03T + 29T^{2} \)
31 \( 1 + 0.867T + 31T^{2} \)
37 \( 1 + 4.68iT - 37T^{2} \)
41 \( 1 - 4.69T + 41T^{2} \)
43 \( 1 - 9.08iT - 43T^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 - 10.9iT - 53T^{2} \)
59 \( 1 + 1.50T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 - 1.68iT - 67T^{2} \)
71 \( 1 + 5.36T + 71T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + 7.39T + 79T^{2} \)
83 \( 1 + 15.3iT - 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + 14.2iT - 97T^{2} \)
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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

−13.43328066578104644410033770787, −12.67401983237172766620926483253, −11.64834893280853757917614104370, −10.50529954340013205434132874180, −9.133004245228679715251236756935, −7.88071401379800783246471717877, −7.10757343929921804937348182940, −5.75809769191137980301822491826, −4.43281394188770735752300012128, −2.01104404704374784062611504125, 2.42434417321214455249833993696, 3.67795916865403692210596670047, 5.62158671867276454602802246111, 7.02648434046855001423477703443, 7.910354409967232809781302784689, 9.891986764811980497593573280697, 10.51573330850688863666608871036, 11.15428844220938189577509964854, 12.55249969942358686869557870908, 13.23678892391073234975765321824

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