↑

Properties

Label 2-115-115.114-c2-0-16
Degree $2$
Conductor $115$
Sign $0.0188 + 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.38i·2-s − 3.81i·3-s − 7.49·4-s + (−4.94 − 0.753i)5-s + 12.9·6-s − 7.85·7-s − 11.8i·8-s − 5.56·9-s + (2.55 − 16.7i)10-s − 14.6i·11-s + 28.5i·12-s − 7.02i·13-s − 26.6i·14-s + (−2.87 + 18.8i)15-s + 10.1·16-s − 15.0·17-s + ⋯
L(s)  = 1  + 1.69i·2-s − 1.27i·3-s − 1.87·4-s + (−0.988 − 0.150i)5-s + 2.15·6-s − 1.12·7-s − 1.47i·8-s − 0.617·9-s + (0.255 − 1.67i)10-s − 1.33i·11-s + 2.38i·12-s − 0.540i·13-s − 1.90i·14-s + (−0.191 + 1.25i)15-s + 0.634·16-s − 0.882·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0188 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0188 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.0188 + 0.999i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ 0.0188 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.232405 - 0.228063i\)
\(L(\frac12)\) \(\approx\) \(0.232405 - 0.228063i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.94 + 0.753i)T \)
23 \( 1 + (-3.89 - 22.6i)T \)
good2 \( 1 - 3.38iT - 4T^{2} \)
3 \( 1 + 3.81iT - 9T^{2} \)
7 \( 1 + 7.85T + 49T^{2} \)
11 \( 1 + 14.6iT - 121T^{2} \)
13 \( 1 + 7.02iT - 169T^{2} \)
17 \( 1 + 15.0T + 289T^{2} \)
19 \( 1 - 19.6iT - 361T^{2} \)
29 \( 1 + 36.8T + 841T^{2} \)
31 \( 1 + 12.6T + 961T^{2} \)
37 \( 1 + 1.40T + 1.36e3T^{2} \)
41 \( 1 - 5.88T + 1.68e3T^{2} \)
43 \( 1 - 81.8T + 1.84e3T^{2} \)
47 \( 1 + 65.3iT - 2.20e3T^{2} \)
53 \( 1 + 59.2T + 2.80e3T^{2} \)
59 \( 1 + 16.2T + 3.48e3T^{2} \)
61 \( 1 + 108. iT - 3.72e3T^{2} \)
67 \( 1 - 95.6T + 4.48e3T^{2} \)
71 \( 1 + 89.4T + 5.04e3T^{2} \)
73 \( 1 + 58.6iT - 5.32e3T^{2} \)
79 \( 1 - 42.3iT - 6.24e3T^{2} \)
83 \( 1 - 57.2T + 6.88e3T^{2} \)
89 \( 1 + 21.1iT - 7.92e3T^{2} \)
97 \( 1 + 101.T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.21428630740534154279202470162, −12.58512599710953758101911765360, −11.18865431903074920158510964034, −9.237671421221858395894379667220, −8.167199211911934311493789378081, −7.48120019821432839102121284425, −6.52707451229641754696706955405, −5.65327388461751094230208393631, −3.65293027422475744441080786420, −0.22857934454341277036846172162, 2.72762797908104553711869693298, 4.03607214158833608585879587813, 4.56847201714175177316264010392, 7.02107294849413854674675538326, 9.098310677063081006833652108610, 9.516719069541388961200560660677, 10.63532845548585713025081735292, 11.21602004953318357689388708916, 12.42307637131625332625022617140, 13.02897704162154350796422159893

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