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Properties

Label 2-110-55.14-c1-0-0
Degree $2$
Conductor $110$
Sign $0.644 - 0.764i$
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  + (−0.587 − 0.809i)2-s + (−1.53 + 0.5i)3-s + (−0.309 + 0.951i)4-s + (−0.829 + 2.07i)5-s + (1.30 + 0.951i)6-s + (4.76 + 1.54i)7-s + (0.951 − 0.309i)8-s + (−0.309 + 0.224i)9-s + (2.16 − 0.549i)10-s + (−0.969 + 3.17i)11-s − 1.61i·12-s + (−1.37 − 1.88i)13-s + (−1.54 − 4.76i)14-s + (0.238 − 3.61i)15-s + (−0.809 − 0.587i)16-s + (−0.0359 + 0.0494i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−0.888 + 0.288i)3-s + (−0.154 + 0.475i)4-s + (−0.370 + 0.928i)5-s + (0.534 + 0.388i)6-s + (1.80 + 0.585i)7-s + (0.336 − 0.109i)8-s + (−0.103 + 0.0748i)9-s + (0.685 − 0.173i)10-s + (−0.292 + 0.956i)11-s − 0.467i·12-s + (−0.380 − 0.523i)13-s + (−0.414 − 1.27i)14-s + (0.0615 − 0.932i)15-s + (−0.202 − 0.146i)16-s + (−0.00871 + 0.0119i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.644 - 0.764i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 0.644 - 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.587154 + 0.272969i\)
\(L(\frac12)\) \(\approx\) \(0.587154 + 0.272969i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 + (0.829 - 2.07i)T \)
11 \( 1 + (0.969 - 3.17i)T \)
good3 \( 1 + (1.53 - 0.5i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-4.76 - 1.54i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.37 + 1.88i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.0359 - 0.0494i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.636 - 1.96i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.91iT - 23T^{2} \)
29 \( 1 + (-1.27 + 3.93i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.18 + 5.21i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (4.33 + 1.40i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.41 + 4.35i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.61iT - 43T^{2} \)
47 \( 1 + (-8.44 + 2.74i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.30 + 1.79i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.599 - 1.84i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.84 - 5.69i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.49iT - 67T^{2} \)
71 \( 1 + (7.13 + 5.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.13 - 1.66i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.38 - 3.91i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.49 + 8.94i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.09T + 89T^{2} \)
97 \( 1 + (4.29 + 5.90i)T + (-29.9 + 92.2i)T^{2} \)
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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

−13.88588659251787177250176888704, −12.08727475813617336021777066981, −11.66197909319977770129544225108, −10.79327079354667277026390447156, −10.00418344854381256549245777600, −8.269064691900721244026862184055, −7.46870483962527157852990769490, −5.61725812561626371819000093724, −4.47076655579873394101440891699, −2.33155104884821459054930317746, 1.02354880150477497830439179101, 4.62721514156444049491008017227, 5.38846688023761903756268233011, 6.89719939484381494742331855215, 8.146252420787678430987263027020, 8.787604572685362048477661973631, 10.61225613828221661546258890146, 11.41645909117206856438818844018, 12.21997077471516701058426598839, 13.72486356113998375125135718637

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