↑

Properties

Label 2-1110-185.117-c1-0-12
Degree $2$
Conductor $1110$
Sign $0.999 + 0.0274i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.707 − 0.707i)3-s + 4-s + (−1.40 + 1.73i)5-s + (−0.707 − 0.707i)6-s + (−3.05 − 3.05i)7-s + 8-s + 1.00i·9-s + (−1.40 + 1.73i)10-s + 3.00i·11-s + (−0.707 − 0.707i)12-s + 6.48·13-s + (−3.05 − 3.05i)14-s + (2.22 − 0.230i)15-s + 16-s + 6.00i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (−0.630 + 0.776i)5-s + (−0.288 − 0.288i)6-s + (−1.15 − 1.15i)7-s + 0.353·8-s + 0.333i·9-s + (−0.445 + 0.548i)10-s + 0.906i·11-s + (−0.204 − 0.204i)12-s + 1.79·13-s + (−0.815 − 0.815i)14-s + (0.574 − 0.0594i)15-s + 0.250·16-s + 1.45i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.999 + 0.0274i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.999 + 0.0274i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.879810451\)
\(L(\frac12)\) \(\approx\) \(1.879810451\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (1.40 - 1.73i)T \)
37 \( 1 + (-5.98 + 1.11i)T \)
good7 \( 1 + (3.05 + 3.05i)T + 7iT^{2} \)
11 \( 1 - 3.00iT - 11T^{2} \)
13 \( 1 - 6.48T + 13T^{2} \)
17 \( 1 - 6.00iT - 17T^{2} \)
19 \( 1 + (-4.44 + 4.44i)T - 19iT^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + (1.51 + 1.51i)T + 29iT^{2} \)
31 \( 1 + (-1.07 + 1.07i)T - 31iT^{2} \)
41 \( 1 + 6.80iT - 41T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 + (-5.31 - 5.31i)T + 47iT^{2} \)
53 \( 1 + (5.63 - 5.63i)T - 53iT^{2} \)
59 \( 1 + (-4.46 + 4.46i)T - 59iT^{2} \)
61 \( 1 + (7.40 - 7.40i)T - 61iT^{2} \)
67 \( 1 + (-2.42 + 2.42i)T - 67iT^{2} \)
71 \( 1 + 9.18T + 71T^{2} \)
73 \( 1 + (-2.01 - 2.01i)T + 73iT^{2} \)
79 \( 1 + (6.59 - 6.59i)T - 79iT^{2} \)
83 \( 1 + (0.481 - 0.481i)T - 83iT^{2} \)
89 \( 1 + (4.35 + 4.35i)T + 89iT^{2} \)
97 \( 1 - 16.2iT - 97T^{2} \)
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

−10.18477726589752450778443070097, −9.068220702009316240254407346551, −7.70072397376957960339871606968, −7.21268216836933200291705006639, −6.44633570061311749743534166791, −5.88166040658451390324982661246, −4.29454061004507181698818758091, −3.78182136926960798681436789622, −2.79074052943085333744914378386, −1.04481829250636095321745967515, 0.956264076849988297587387589005, 3.05512322446990610158729627621, 3.52231317088512780338158649900, 4.70264126830839938523839783641, 5.80393535925644282794067874206, 5.91816617780035317723459572029, 7.19245202234501728233590020513, 8.368096174551870919476543032576, 9.060943220456158234817628440368, 9.702343706195158605757180156651

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