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Properties

Label 2-114-57.2-c1-0-2
Degree 22
Conductor 114114
Sign 0.930+0.365i0.930 + 0.365i
Analytic cond. 0.9102940.910294
Root an. cond. 0.9540930.954093
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  + (0.173 − 0.984i)2-s + (1.67 + 0.431i)3-s + (−0.939 − 0.342i)4-s + (1.14 + 3.13i)5-s + (0.716 − 1.57i)6-s + (−1.07 − 1.85i)7-s + (−0.5 + 0.866i)8-s + (2.62 + 1.44i)9-s + (3.28 − 0.579i)10-s + (−5.41 − 3.12i)11-s + (−1.42 − 0.979i)12-s + (−2.56 − 3.05i)13-s + (−2.01 + 0.734i)14-s + (0.560 + 5.75i)15-s + (0.766 + 0.642i)16-s + (−0.403 − 0.0711i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (0.968 + 0.249i)3-s + (−0.469 − 0.171i)4-s + (0.510 + 1.40i)5-s + (0.292 − 0.643i)6-s + (−0.405 − 0.702i)7-s + (−0.176 + 0.306i)8-s + (0.875 + 0.482i)9-s + (1.03 − 0.183i)10-s + (−1.63 − 0.943i)11-s + (−0.412 − 0.282i)12-s + (−0.710 − 0.846i)13-s + (−0.539 + 0.196i)14-s + (0.144 + 1.48i)15-s + (0.191 + 0.160i)16-s + (−0.0978 − 0.0172i)17-s + ⋯

Functional equation

Λ(s)=(114s/2 ΓC(s) L(s)=((0.930+0.365i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(114s/2 ΓC(s+1/2) L(s)=((0.930+0.365i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 114114    =    2⋅3⋅192 \cdot 3 \cdot 19
Sign: 0.930+0.365i0.930 + 0.365i
Analytic conductor: 0.9102940.910294
Root analytic conductor: 0.9540930.954093
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ114(59,⋅)\chi_{114} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 114, ( :1/2), 0.930+0.365i)(2,\ 114,\ (\ :1/2),\ 0.930 + 0.365i)

Particular Values

L(1)L(1) ≈\approx 1.33342−0.252193i1.33342 - 0.252193i
L(12)L(\frac12) ≈\approx 1.33342−0.252193i1.33342 - 0.252193i
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+(−0.173+0.984i)T 1 + (-0.173 + 0.984i)T
3 1+(−1.67−0.431i)T 1 + (-1.67 - 0.431i)T
19 1+(−4.34+0.329i)T 1 + (-4.34 + 0.329i)T
good5 1+(−1.14−3.13i)T+(−3.83+3.21i)T2 1 + (-1.14 - 3.13i)T + (-3.83 + 3.21i)T^{2}
7 1+(1.07+1.85i)T+(−3.5+6.06i)T2 1 + (1.07 + 1.85i)T + (-3.5 + 6.06i)T^{2}
11 1+(5.41+3.12i)T+(5.5+9.52i)T2 1 + (5.41 + 3.12i)T + (5.5 + 9.52i)T^{2}
13 1+(2.56+3.05i)T+(−2.25+12.8i)T2 1 + (2.56 + 3.05i)T + (-2.25 + 12.8i)T^{2}
17 1+(0.403+0.0711i)T+(15.9+5.81i)T2 1 + (0.403 + 0.0711i)T + (15.9 + 5.81i)T^{2}
23 1+(0.280−0.770i)T+(−17.6−14.7i)T2 1 + (0.280 - 0.770i)T + (-17.6 - 14.7i)T^{2}
29 1+(−0.805−4.56i)T+(−27.2+9.91i)T2 1 + (-0.805 - 4.56i)T + (-27.2 + 9.91i)T^{2}
31 1+(2.02−1.16i)T+(15.5−26.8i)T2 1 + (2.02 - 1.16i)T + (15.5 - 26.8i)T^{2}
37 1−6.01iT−37T2 1 - 6.01iT - 37T^{2}
41 1+(−0.926−0.777i)T+(7.11+40.3i)T2 1 + (-0.926 - 0.777i)T + (7.11 + 40.3i)T^{2}
43 1+(−5.87+2.13i)T+(32.9−27.6i)T2 1 + (-5.87 + 2.13i)T + (32.9 - 27.6i)T^{2}
47 1+(7.59−1.33i)T+(44.1−16.0i)T2 1 + (7.59 - 1.33i)T + (44.1 - 16.0i)T^{2}
53 1+(0.220+0.0802i)T+(40.6+34.0i)T2 1 + (0.220 + 0.0802i)T + (40.6 + 34.0i)T^{2}
59 1+(−0.930+5.27i)T+(−55.4−20.1i)T2 1 + (-0.930 + 5.27i)T + (-55.4 - 20.1i)T^{2}
61 1+(−7.30−2.65i)T+(46.7+39.2i)T2 1 + (-7.30 - 2.65i)T + (46.7 + 39.2i)T^{2}
67 1+(−3.48+0.614i)T+(62.9−22.9i)T2 1 + (-3.48 + 0.614i)T + (62.9 - 22.9i)T^{2}
71 1+(−4.19+1.52i)T+(54.3−45.6i)T2 1 + (-4.19 + 1.52i)T + (54.3 - 45.6i)T^{2}
73 1+(4.33+3.63i)T+(12.6+71.8i)T2 1 + (4.33 + 3.63i)T + (12.6 + 71.8i)T^{2}
79 1+(8.05−9.59i)T+(−13.7−77.7i)T2 1 + (8.05 - 9.59i)T + (-13.7 - 77.7i)T^{2}
83 1+(−8.01+4.62i)T+(41.5−71.8i)T2 1 + (-8.01 + 4.62i)T + (41.5 - 71.8i)T^{2}
89 1+(5.61−4.71i)T+(15.4−87.6i)T2 1 + (5.61 - 4.71i)T + (15.4 - 87.6i)T^{2}
97 1+(16.0+2.83i)T+(91.1+33.1i)T2 1 + (16.0 + 2.83i)T + (91.1 + 33.1i)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

−13.60592128396102487601644051938, −12.85776320182177351781182275783, −11.01163731194803310900725370081, −10.34380500809456920032566337423, −9.745670671319601744156541285195, −8.111503613329624352762278843883, −7.09578501697515224646991153925, −5.31935155958272448085992797205, −3.32136686441162803471039806262, −2.71487538198983525595867719210, 2.34059615826681349755095999868, 4.53584581617290505220326034315, 5.57881508231541731927475100820, 7.25975451281761104190035676622, 8.224617009009099915197045550490, 9.310028009968613094214829893903, 9.794302412235428772467292120253, 12.24339507368346136117161501361, 12.83435064254517825779387854396, 13.56071282024081886167599116633

Graph of the ZZ-function along the critical line