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Properties

Label 2-110-11.3-c3-0-2
Degree $2$
Conductor $110$
Sign $0.967 - 0.254i$
Analytic cond. $6.49021$
Root an. cond. $2.54758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 + 1.17i)2-s + (−0.678 − 2.08i)3-s + (1.23 − 3.80i)4-s + (−4.04 − 2.93i)5-s + (3.55 + 2.57i)6-s + (−10.1 + 31.2i)7-s + (2.47 + 7.60i)8-s + (17.9 − 13.0i)9-s + 10·10-s + (22.4 − 28.7i)11-s − 8.77·12-s + (52.1 − 37.9i)13-s + (−20.2 − 62.4i)14-s + (−3.39 + 10.4i)15-s + (−12.9 − 9.40i)16-s + (77.1 + 56.0i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.130 − 0.401i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.241 + 0.175i)6-s + (−0.548 + 1.68i)7-s + (0.109 + 0.336i)8-s + (0.664 − 0.482i)9-s + 0.316·10-s + (0.614 − 0.789i)11-s − 0.211·12-s + (1.11 − 0.808i)13-s + (−0.387 − 1.19i)14-s + (−0.0583 + 0.179i)15-s + (−0.202 − 0.146i)16-s + (1.10 + 0.799i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.967 - 0.254i$
Analytic conductor: \(6.49021\)
Root analytic conductor: \(2.54758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :3/2),\ 0.967 - 0.254i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.14394 + 0.147990i\)
\(L(\frac12)\) \(\approx\) \(1.14394 + 0.147990i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.61 - 1.17i)T \)
5 \( 1 + (4.04 + 2.93i)T \)
11 \( 1 + (-22.4 + 28.7i)T \)
good3 \( 1 + (0.678 + 2.08i)T + (-21.8 + 15.8i)T^{2} \)
7 \( 1 + (10.1 - 31.2i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (-52.1 + 37.9i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-77.1 - 56.0i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (-21.2 - 65.5i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 - 112.T + 1.21e4T^{2} \)
29 \( 1 + (-18.1 + 55.9i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (6.31 - 4.58i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-34.2 + 105. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (-42.2 - 130. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 + 23.7T + 7.95e4T^{2} \)
47 \( 1 + (-167. - 516. i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (38.8 - 28.2i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-272. + 838. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (339. + 246. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + 726.T + 3.00e5T^{2} \)
71 \( 1 + (-921. - 669. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (41.9 - 129. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (333. - 242. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (110. + 80.0i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 + 492.T + 7.04e5T^{2} \)
97 \( 1 + (-997. + 724. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−12.91893991173974483541195756400, −12.29753368525829730120115584158, −11.19668386422021428371443663918, −9.735741645913600794333095673565, −8.782516299042087027574010116644, −7.930249942623643887564334350281, −6.30124994075559795456040057725, −5.70283208974898972650823755862, −3.39828311370164772762037430070, −1.15582733606615493691160718329, 1.10419481947305048895514805951, 3.52294502204010396492092696932, 4.50614143513939556355980674303, 6.92303447521141865557182081931, 7.39261507406241713899196303270, 9.133533383915780969319828956146, 10.10509277792402678731081168656, 10.80608739340610524252432969907, 11.80993739387766166673700938474, 13.18747695674494460112669370024

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