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Properties

Label 2-115-5.3-c2-0-11
Degree $2$
Conductor $115$
Sign $-0.0679 + 0.997i$
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  + (−2.11 + 2.11i)2-s + (−0.751 − 0.751i)3-s − 4.91i·4-s + (−1.47 + 4.77i)5-s + 3.17·6-s + (−3.30 + 3.30i)7-s + (1.92 + 1.92i)8-s − 7.87i·9-s + (−6.96 − 13.2i)10-s − 2.07·11-s + (−3.69 + 3.69i)12-s + (−9.71 − 9.71i)13-s − 13.9i·14-s + (4.70 − 2.48i)15-s + 11.5·16-s + (−2.26 + 2.26i)17-s + ⋯
L(s)  = 1  + (−1.05 + 1.05i)2-s + (−0.250 − 0.250i)3-s − 1.22i·4-s + (−0.295 + 0.955i)5-s + 0.528·6-s + (−0.472 + 0.472i)7-s + (0.240 + 0.240i)8-s − 0.874i·9-s + (−0.696 − 1.32i)10-s − 0.188·11-s + (−0.307 + 0.307i)12-s + (−0.747 − 0.747i)13-s − 0.997i·14-s + (0.313 − 0.165i)15-s + 0.720·16-s + (−0.133 + 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0679 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0679 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.0679 + 0.997i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ -0.0679 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0453026 - 0.0484911i\)
\(L(\frac12)\) \(\approx\) \(0.0453026 - 0.0484911i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.47 - 4.77i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good2 \( 1 + (2.11 - 2.11i)T - 4iT^{2} \)
3 \( 1 + (0.751 + 0.751i)T + 9iT^{2} \)
7 \( 1 + (3.30 - 3.30i)T - 49iT^{2} \)
11 \( 1 + 2.07T + 121T^{2} \)
13 \( 1 + (9.71 + 9.71i)T + 169iT^{2} \)
17 \( 1 + (2.26 - 2.26i)T - 289iT^{2} \)
19 \( 1 + 32.3iT - 361T^{2} \)
29 \( 1 - 21.3iT - 841T^{2} \)
31 \( 1 + 47.1T + 961T^{2} \)
37 \( 1 + (30.9 - 30.9i)T - 1.36e3iT^{2} \)
41 \( 1 + 8.65T + 1.68e3T^{2} \)
43 \( 1 + (23.0 + 23.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-4.91 + 4.91i)T - 2.20e3iT^{2} \)
53 \( 1 + (-16.0 - 16.0i)T + 2.80e3iT^{2} \)
59 \( 1 - 15.8iT - 3.48e3T^{2} \)
61 \( 1 - 92.5T + 3.72e3T^{2} \)
67 \( 1 + (-2.96 + 2.96i)T - 4.48e3iT^{2} \)
71 \( 1 + 112.T + 5.04e3T^{2} \)
73 \( 1 + (-4.53 - 4.53i)T + 5.32e3iT^{2} \)
79 \( 1 + 17.0iT - 6.24e3T^{2} \)
83 \( 1 + (76.6 + 76.6i)T + 6.88e3iT^{2} \)
89 \( 1 - 103. iT - 7.92e3T^{2} \)
97 \( 1 + (-71.0 + 71.0i)T - 9.40e3iT^{2} \)
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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

−13.02084005861265448120258657526, −11.95113123213863465426401468239, −10.68135255018502043752566413433, −9.606016605947787981348976484163, −8.686000090639929884506072245942, −7.26679601011140317337428741826, −6.79213666930022439111352052226, −5.58719811838846120155166116382, −3.13136850485934591586914615576, −0.06592429037849532802622447156, 1.89194620219527878244810521655, 3.91532842591014316289147973827, 5.41756216534105635030705367609, 7.49537016278646516629779532102, 8.499514791334228870636778297998, 9.617087453940746871898052941015, 10.30588386308642189305102098397, 11.38793066681877479480364704789, 12.24998737600441809343677314895, 13.16466211480802301496731520471

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