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Properties

Label 2-115-5.2-c2-0-7
Degree $2$
Conductor $115$
Sign $0.813 + 0.582i$
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  + (−1.95 − 1.95i)2-s + (−0.210 + 0.210i)3-s + 3.61i·4-s + (4.62 + 1.89i)5-s + 0.820·6-s + (2.37 + 2.37i)7-s + (−0.749 + 0.749i)8-s + 8.91i·9-s + (−5.31 − 12.7i)10-s + 12.6·11-s + (−0.760 − 0.760i)12-s + (10.8 − 10.8i)13-s − 9.25i·14-s + (−1.37 + 0.573i)15-s + 17.3·16-s + (−15.9 − 15.9i)17-s + ⋯
L(s)  = 1  + (−0.975 − 0.975i)2-s + (−0.0700 + 0.0700i)3-s + 0.904i·4-s + (0.925 + 0.379i)5-s + 0.136·6-s + (0.338 + 0.338i)7-s + (−0.0936 + 0.0936i)8-s + 0.990i·9-s + (−0.531 − 1.27i)10-s + 1.15·11-s + (−0.0633 − 0.0633i)12-s + (0.835 − 0.835i)13-s − 0.661i·14-s + (−0.0914 + 0.0382i)15-s + 1.08·16-s + (−0.940 − 0.940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.813 + 0.582i$
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: \((2,\ 115,\ (\ :1),\ 0.813 + 0.582i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.956664 - 0.307203i\)
\(L(\frac12)\) \(\approx\) \(0.956664 - 0.307203i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.62 - 1.89i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good2 \( 1 + (1.95 + 1.95i)T + 4iT^{2} \)
3 \( 1 + (0.210 - 0.210i)T - 9iT^{2} \)
7 \( 1 + (-2.37 - 2.37i)T + 49iT^{2} \)
11 \( 1 - 12.6T + 121T^{2} \)
13 \( 1 + (-10.8 + 10.8i)T - 169iT^{2} \)
17 \( 1 + (15.9 + 15.9i)T + 289iT^{2} \)
19 \( 1 + 0.760iT - 361T^{2} \)
29 \( 1 - 42.2iT - 841T^{2} \)
31 \( 1 - 37.1T + 961T^{2} \)
37 \( 1 + (-33.8 - 33.8i)T + 1.36e3iT^{2} \)
41 \( 1 + 33.6T + 1.68e3T^{2} \)
43 \( 1 + (-5.27 + 5.27i)T - 1.84e3iT^{2} \)
47 \( 1 + (57.2 + 57.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (36.4 - 36.4i)T - 2.80e3iT^{2} \)
59 \( 1 + 22.9iT - 3.48e3T^{2} \)
61 \( 1 + 89.6T + 3.72e3T^{2} \)
67 \( 1 + (-54.0 - 54.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 67.9T + 5.04e3T^{2} \)
73 \( 1 + (10.7 - 10.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 92.3iT - 6.24e3T^{2} \)
83 \( 1 + (-7.05 + 7.05i)T - 6.88e3iT^{2} \)
89 \( 1 + 27.4iT - 7.92e3T^{2} \)
97 \( 1 + (27.7 + 27.7i)T + 9.40e3iT^{2} \)
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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

−13.14546073110867058303256045391, −11.71667142597259528229618209247, −10.97845851629786848924560299417, −10.16281019502451573087521227197, −9.143034006398008175225180262071, −8.285267362075964106796574322573, −6.56726192663553969405050752666, −5.15355806885957031895922207546, −2.89494194080091155416385096272, −1.54379055910652790973886531815, 1.27220418352267477200377503412, 4.13046505971364195242113765929, 6.32379681899621687667010493947, 6.42758063035731230439970773257, 8.171698007528709734773593771653, 9.127299310137227944041397550907, 9.679357440438611901388209441351, 11.18806887185687531441196914404, 12.43423235933009774611417788551, 13.65370827974727382569665790912

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