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Properties

Label 2-110-11.5-c1-0-0
Degree $2$
Conductor $110$
Sign $0.263 - 0.964i$
Analytic cond. $0.878354$
Root an. cond. $0.937205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (1 + 0.726i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.381 + 1.17i)6-s + (−0.309 + 0.224i)7-s + (−0.809 − 0.587i)8-s + (−0.454 − 1.40i)9-s − 0.999·10-s + (3.04 − 1.31i)11-s − 1.23·12-s + (0.190 + 0.587i)13-s + (−0.309 − 0.224i)14-s + (−1 + 0.726i)15-s + (0.309 − 0.951i)16-s + (−0.236 + 0.726i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.577 + 0.419i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (−0.155 + 0.479i)6-s + (−0.116 + 0.0848i)7-s + (−0.286 − 0.207i)8-s + (−0.151 − 0.466i)9-s − 0.316·10-s + (0.918 − 0.396i)11-s − 0.356·12-s + (0.0529 + 0.163i)13-s + (−0.0825 − 0.0600i)14-s + (−0.258 + 0.187i)15-s + (0.0772 − 0.237i)16-s + (−0.0572 + 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.263 - 0.964i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 0.263 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.994715 + 0.759476i\)
\(L(\frac12)\) \(\approx\) \(0.994715 + 0.759476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-3.04 + 1.31i)T \)
good3 \( 1 + (-1 - 0.726i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (0.309 - 0.224i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.190 - 0.587i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.236 - 0.726i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.73 + 4.16i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.85T + 23T^{2} \)
29 \( 1 + (2.61 - 1.90i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.854 + 2.62i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.73 - 3.44i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.92 + 2.85i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 + (-3.5 - 2.54i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.73 - 11.4i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.73 - 2.71i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.76 - 8.50i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.23T + 67T^{2} \)
71 \( 1 + (-2.70 + 8.33i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (9.47 - 6.88i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.23 + 9.95i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.708 + 2.17i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + (3.70 + 11.4i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.15324269718339693571359835983, −13.12902346154607485291213849851, −11.87510989411087655579837240890, −10.69053749783921219598275728286, −9.181592322698270081709106631142, −8.689692898141300890779335872969, −7.09085273254919598748690297912, −6.15279257623887221919274836612, −4.38616943913952582745886273440, −3.16386272298179551894020807461, 1.88024641050120290946512859784, 3.63118161087949276640279258260, 5.08441051044528265069281701914, 6.79689793270003352740763166708, 8.234077072523460416645200531643, 9.095124768403709638293399825351, 10.36571270774770259694141174000, 11.48934099653248746502709963391, 12.60097075157440624879704782671, 13.24689497388653958260232538794

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