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Properties

Label 2-110-11.4-c1-0-3
Degree 22
Conductor 110110
Sign −0.220+0.975i-0.220 + 0.975i
Analytic cond. 0.8783540.878354
Root an. cond. 0.9372050.937205
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (1 − 3.07i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (−2.61 + 1.90i)6-s + (0.809 + 2.48i)7-s + (0.309 − 0.951i)8-s + (−6.04 − 4.39i)9-s − 10-s + (−2.54 + 2.12i)11-s + 3.23·12-s + (1.30 + 0.951i)13-s + (0.809 − 2.48i)14-s + (−1 − 3.07i)15-s + (−0.809 + 0.587i)16-s + (4.23 − 3.07i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.577 − 1.77i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (−1.06 + 0.776i)6-s + (0.305 + 0.941i)7-s + (0.109 − 0.336i)8-s + (−2.01 − 1.46i)9-s − 0.316·10-s + (−0.767 + 0.641i)11-s + 0.934·12-s + (0.363 + 0.263i)13-s + (0.216 − 0.665i)14-s + (−0.258 − 0.794i)15-s + (−0.202 + 0.146i)16-s + (1.02 − 0.746i)17-s + ⋯

Functional equation

Λ(s)=(110s/2 ΓC(s) L(s)=((−0.220+0.975i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(110s/2 ΓC(s+1/2) L(s)=((−0.220+0.975i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110110    =    2⋅5⋅112 \cdot 5 \cdot 11
Sign: −0.220+0.975i-0.220 + 0.975i
Analytic conductor: 0.8783540.878354
Root analytic conductor: 0.9372050.937205
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ110(81,⋅)\chi_{110} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 110, ( :1/2), −0.220+0.975i)(2,\ 110,\ (\ :1/2),\ -0.220 + 0.975i)

Particular Values

L(1)L(1) ≈\approx 0.603531−0.755440i0.603531 - 0.755440i
L(12)L(\frac12) ≈\approx 0.603531−0.755440i0.603531 - 0.755440i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
5 1+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
11 1+(2.54−2.12i)T 1 + (2.54 - 2.12i)T
good3 1+(−1+3.07i)T+(−2.42−1.76i)T2 1 + (-1 + 3.07i)T + (-2.42 - 1.76i)T^{2}
7 1+(−0.809−2.48i)T+(−5.66+4.11i)T2 1 + (-0.809 - 2.48i)T + (-5.66 + 4.11i)T^{2}
13 1+(−1.30−0.951i)T+(4.01+12.3i)T2 1 + (-1.30 - 0.951i)T + (4.01 + 12.3i)T^{2}
17 1+(−4.23+3.07i)T+(5.25−16.1i)T2 1 + (-4.23 + 3.07i)T + (5.25 - 16.1i)T^{2}
19 1+(1.26−3.88i)T+(−15.3−11.1i)T2 1 + (1.26 - 3.88i)T + (-15.3 - 11.1i)T^{2}
23 1−0.145T+23T2 1 - 0.145T + 23T^{2}
29 1+(0.381+1.17i)T+(−23.4+17.0i)T2 1 + (0.381 + 1.17i)T + (-23.4 + 17.0i)T^{2}
31 1+(−5.85−4.25i)T+(9.57+29.4i)T2 1 + (-5.85 - 4.25i)T + (9.57 + 29.4i)T^{2}
37 1+(0.263+0.812i)T+(−29.9+21.7i)T2 1 + (0.263 + 0.812i)T + (-29.9 + 21.7i)T^{2}
41 1+(0.572−1.76i)T+(−33.1−24.0i)T2 1 + (0.572 - 1.76i)T + (-33.1 - 24.0i)T^{2}
43 1+9.23T+43T2 1 + 9.23T + 43T^{2}
47 1+(−3.5+10.7i)T+(−38.0−27.6i)T2 1 + (-3.5 + 10.7i)T + (-38.0 - 27.6i)T^{2}
53 1+(0.736+0.534i)T+(16.3+50.4i)T2 1 + (0.736 + 0.534i)T + (16.3 + 50.4i)T^{2}
59 1+(−0.736−2.26i)T+(−47.7+34.6i)T2 1 + (-0.736 - 2.26i)T + (-47.7 + 34.6i)T^{2}
61 1+(7.23−5.25i)T+(18.8−58.0i)T2 1 + (7.23 - 5.25i)T + (18.8 - 58.0i)T^{2}
67 1−0.763T+67T2 1 - 0.763T + 67T^{2}
71 1+(10.7−7.77i)T+(21.9−67.5i)T2 1 + (10.7 - 7.77i)T + (21.9 - 67.5i)T^{2}
73 1+(0.527+1.62i)T+(−59.0+42.9i)T2 1 + (0.527 + 1.62i)T + (-59.0 + 42.9i)T^{2}
79 1+(−1.23−0.898i)T+(24.4+75.1i)T2 1 + (-1.23 - 0.898i)T + (24.4 + 75.1i)T^{2}
83 1+(12.7−9.23i)T+(25.6−78.9i)T2 1 + (12.7 - 9.23i)T + (25.6 - 78.9i)T^{2}
89 1+12.0T+89T2 1 + 12.0T + 89T^{2}
97 1+(−9.70−7.05i)T+(29.9+92.2i)T2 1 + (-9.70 - 7.05i)T + (29.9 + 92.2i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−13.18023162403324771957452460501, −12.24473314173808180428992358915, −11.79382528359755014480301140282, −10.01163188693441557956723195924, −8.725274363632144712685088363542, −8.070063971714528121583775495351, −6.98091356769265814758596819409, −5.62095413985784400899036655819, −2.79218766861194229476159210542, −1.63120457434034122612308164467, 3.14377541079798043747284821099, 4.60448263139425405692379786559, 5.87165781259701693233645598767, 7.78732002411149714953887795002, 8.662980336039716331505079858087, 9.850950899887139770989817838847, 10.50779665986759223933459870856, 11.13612388410584567383626327300, 13.46365730799103962294448377100, 14.24129415487598630950886074540

Graph of the ZZ-function along the critical line