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Properties

Label 2-115-5.2-c2-0-8
Degree $2$
Conductor $115$
Sign $-0.819 - 0.573i$
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  + (2.29 + 2.29i)2-s + (−2.60 + 2.60i)3-s + 6.55i·4-s + (4.64 + 1.84i)5-s − 11.9·6-s + (−4.55 − 4.55i)7-s + (−5.88 + 5.88i)8-s − 4.54i·9-s + (6.42 + 14.9i)10-s + 0.500·11-s + (−17.0 − 17.0i)12-s + (6.17 − 6.17i)13-s − 20.9i·14-s + (−16.8 + 7.27i)15-s − 0.789·16-s + (1.49 + 1.49i)17-s + ⋯
L(s)  = 1  + (1.14 + 1.14i)2-s + (−0.867 + 0.867i)3-s + 1.63i·4-s + (0.929 + 0.369i)5-s − 1.99·6-s + (−0.650 − 0.650i)7-s + (−0.735 + 0.735i)8-s − 0.504i·9-s + (0.642 + 1.49i)10-s + 0.0455·11-s + (−1.42 − 1.42i)12-s + (0.475 − 0.475i)13-s − 1.49i·14-s + (−1.12 + 0.485i)15-s − 0.0493·16-s + (0.0876 + 0.0876i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.819 - 0.573i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ -0.819 - 0.573i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.605573 + 1.92156i\)
\(L(\frac12)\) \(\approx\) \(0.605573 + 1.92156i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.64 - 1.84i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good2 \( 1 + (-2.29 - 2.29i)T + 4iT^{2} \)
3 \( 1 + (2.60 - 2.60i)T - 9iT^{2} \)
7 \( 1 + (4.55 + 4.55i)T + 49iT^{2} \)
11 \( 1 - 0.500T + 121T^{2} \)
13 \( 1 + (-6.17 + 6.17i)T - 169iT^{2} \)
17 \( 1 + (-1.49 - 1.49i)T + 289iT^{2} \)
19 \( 1 - 21.0iT - 361T^{2} \)
29 \( 1 + 43.1iT - 841T^{2} \)
31 \( 1 - 57.0T + 961T^{2} \)
37 \( 1 + (13.6 + 13.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 31.0T + 1.68e3T^{2} \)
43 \( 1 + (40.9 - 40.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (60.7 + 60.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-0.965 + 0.965i)T - 2.80e3iT^{2} \)
59 \( 1 + 1.59iT - 3.48e3T^{2} \)
61 \( 1 + 14.5T + 3.72e3T^{2} \)
67 \( 1 + (91.2 + 91.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 72.1T + 5.04e3T^{2} \)
73 \( 1 + (-59.6 + 59.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 80.4iT - 6.24e3T^{2} \)
83 \( 1 + (45.6 - 45.6i)T - 6.88e3iT^{2} \)
89 \( 1 - 61.2iT - 7.92e3T^{2} \)
97 \( 1 + (-112. - 112. i)T + 9.40e3iT^{2} \)
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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

−13.71752692587925909305624501341, −13.21811333466877564535935779028, −11.86747655485967929579170704089, −10.42398247536629833343811542820, −9.885821292772197936160154802992, −7.908748814163412429453999921872, −6.41416243908149227384811226263, −5.93283312396439288212737163977, −4.76675738186697047168328761327, −3.53937655759855208910174642658, 1.35214992675166087079776376684, 2.84002486938633519663327937026, 4.80972827599538769499566603189, 5.87797397489246305358604258043, 6.65635870945101280477359824442, 8.938028308152661614331890559628, 10.13273594269713157401044574120, 11.30406890598833589371230687248, 12.09177508792894422296116566224, 12.84091301731222931393417299294

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