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Properties

Label 2-114-57.2-c1-0-5
Degree 22
Conductor 114114
Sign −0.184+0.982i-0.184 + 0.982i
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 + (0.517 − 1.65i)3-s + (−0.939 − 0.342i)4-s + (−0.258 − 0.710i)5-s + (−1.53 − 0.797i)6-s + (0.777 + 1.34i)7-s + (−0.5 + 0.866i)8-s + (−2.46 − 1.71i)9-s + (−0.744 + 0.131i)10-s + (0.832 + 0.480i)11-s + (−1.05 + 1.37i)12-s + (0.416 + 0.496i)13-s + (1.46 − 0.532i)14-s + (−1.30 + 0.0594i)15-s + (0.766 + 0.642i)16-s + (6.73 + 1.18i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (0.298 − 0.954i)3-s + (−0.469 − 0.171i)4-s + (−0.115 − 0.317i)5-s + (−0.627 − 0.325i)6-s + (0.294 + 0.509i)7-s + (−0.176 + 0.306i)8-s + (−0.821 − 0.570i)9-s + (−0.235 + 0.0415i)10-s + (0.250 + 0.144i)11-s + (−0.303 + 0.397i)12-s + (0.115 + 0.137i)13-s + (0.390 − 0.142i)14-s + (−0.337 + 0.0153i)15-s + (0.191 + 0.160i)16-s + (1.63 + 0.287i)17-s + ⋯

Functional equation

Λ(s)=(114s/2 ΓC(s) L(s)=((−0.184+0.982i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(114s/2 ΓC(s+1/2) L(s)=((−0.184+0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 114114    =    2⋅3⋅192 \cdot 3 \cdot 19
Sign: −0.184+0.982i-0.184 + 0.982i
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), −0.184+0.982i)(2,\ 114,\ (\ :1/2),\ -0.184 + 0.982i)

Particular Values

L(1)L(1) ≈\approx 0.729462−0.879312i0.729462 - 0.879312i
L(12)L(\frac12) ≈\approx 0.729462−0.879312i0.729462 - 0.879312i
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+(−0.517+1.65i)T 1 + (-0.517 + 1.65i)T
19 1+(4.14−1.35i)T 1 + (4.14 - 1.35i)T
good5 1+(0.258+0.710i)T+(−3.83+3.21i)T2 1 + (0.258 + 0.710i)T + (-3.83 + 3.21i)T^{2}
7 1+(−0.777−1.34i)T+(−3.5+6.06i)T2 1 + (-0.777 - 1.34i)T + (-3.5 + 6.06i)T^{2}
11 1+(−0.832−0.480i)T+(5.5+9.52i)T2 1 + (-0.832 - 0.480i)T + (5.5 + 9.52i)T^{2}
13 1+(−0.416−0.496i)T+(−2.25+12.8i)T2 1 + (-0.416 - 0.496i)T + (-2.25 + 12.8i)T^{2}
17 1+(−6.73−1.18i)T+(15.9+5.81i)T2 1 + (-6.73 - 1.18i)T + (15.9 + 5.81i)T^{2}
23 1+(−0.400+1.10i)T+(−17.6−14.7i)T2 1 + (-0.400 + 1.10i)T + (-17.6 - 14.7i)T^{2}
29 1+(−1.39−7.92i)T+(−27.2+9.91i)T2 1 + (-1.39 - 7.92i)T + (-27.2 + 9.91i)T^{2}
31 1+(2.63−1.52i)T+(15.5−26.8i)T2 1 + (2.63 - 1.52i)T + (15.5 - 26.8i)T^{2}
37 1−4.12iT−37T2 1 - 4.12iT - 37T^{2}
41 1+(4.09+3.43i)T+(7.11+40.3i)T2 1 + (4.09 + 3.43i)T + (7.11 + 40.3i)T^{2}
43 1+(7.34−2.67i)T+(32.9−27.6i)T2 1 + (7.34 - 2.67i)T + (32.9 - 27.6i)T^{2}
47 1+(3.11−0.548i)T+(44.1−16.0i)T2 1 + (3.11 - 0.548i)T + (44.1 - 16.0i)T^{2}
53 1+(13.6+4.96i)T+(40.6+34.0i)T2 1 + (13.6 + 4.96i)T + (40.6 + 34.0i)T^{2}
59 1+(−2.02+11.4i)T+(−55.4−20.1i)T2 1 + (-2.02 + 11.4i)T + (-55.4 - 20.1i)T^{2}
61 1+(−10.1−3.70i)T+(46.7+39.2i)T2 1 + (-10.1 - 3.70i)T + (46.7 + 39.2i)T^{2}
67 1+(9.19−1.62i)T+(62.9−22.9i)T2 1 + (9.19 - 1.62i)T + (62.9 - 22.9i)T^{2}
71 1+(−0.0322+0.0117i)T+(54.3−45.6i)T2 1 + (-0.0322 + 0.0117i)T + (54.3 - 45.6i)T^{2}
73 1+(3.04+2.55i)T+(12.6+71.8i)T2 1 + (3.04 + 2.55i)T + (12.6 + 71.8i)T^{2}
79 1+(0.893−1.06i)T+(−13.7−77.7i)T2 1 + (0.893 - 1.06i)T + (-13.7 - 77.7i)T^{2}
83 1+(10.4−6.05i)T+(41.5−71.8i)T2 1 + (10.4 - 6.05i)T + (41.5 - 71.8i)T^{2}
89 1+(−4.68+3.92i)T+(15.4−87.6i)T2 1 + (-4.68 + 3.92i)T + (15.4 - 87.6i)T^{2}
97 1+(−9.54−1.68i)T+(91.1+33.1i)T2 1 + (-9.54 - 1.68i)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

−12.94017169370280250183319474119, −12.38834027264263289499241841151, −11.56291898639289080631777791898, −10.22804778511427163997357574909, −8.837972739532904457400059254134, −8.120530073209348129558961756331, −6.57767324312862377559979415772, −5.16396058582476973217693566769, −3.31321013956548319010295501370, −1.61950410827600855092488691341, 3.34483724372996788864475836307, 4.58817094953926010843683071487, 5.87594332519679508753761771036, 7.41357765521036412336681041048, 8.416021669883106392291555840484, 9.610172311796264846283338001059, 10.57030123123334137049116453992, 11.67083358712823674657957585571, 13.19809185373099852823073772677, 14.25313797082788886889186664879

Graph of the ZZ-function along the critical line