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Properties

Label 2-15e2-225.106-c1-0-12
Degree 22
Conductor 225225
Sign 0.933+0.358i0.933 + 0.358i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
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  + (0.199 + 0.0424i)2-s + (−1.41 + 0.998i)3-s + (−1.78 − 0.796i)4-s + (2.05 − 0.891i)5-s + (−0.325 + 0.139i)6-s + (0.530 − 0.918i)7-s + (−0.654 − 0.475i)8-s + (1.00 − 2.82i)9-s + (0.447 − 0.0909i)10-s + (5.73 + 1.21i)11-s + (3.32 − 0.658i)12-s + (3.20 − 0.680i)13-s + (0.144 − 0.160i)14-s + (−2.01 + 3.30i)15-s + (2.51 + 2.78i)16-s + (−5.93 − 4.31i)17-s + ⋯
L(s)  = 1  + (0.141 + 0.0300i)2-s + (−0.817 + 0.576i)3-s + (−0.894 − 0.398i)4-s + (0.917 − 0.398i)5-s + (−0.132 + 0.0568i)6-s + (0.200 − 0.346i)7-s + (−0.231 − 0.168i)8-s + (0.335 − 0.941i)9-s + (0.141 − 0.0287i)10-s + (1.72 + 0.367i)11-s + (0.960 − 0.190i)12-s + (0.887 − 0.188i)13-s + (0.0387 − 0.0430i)14-s + (−0.519 + 0.854i)15-s + (0.627 + 0.696i)16-s + (−1.43 − 1.04i)17-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((0.933+0.358i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((0.933+0.358i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: 0.933+0.358i0.933 + 0.358i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ225(106,⋅)\chi_{225} (106, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :1/2), 0.933+0.358i)(2,\ 225,\ (\ :1/2),\ 0.933 + 0.358i)

Particular Values

L(1)L(1) ≈\approx 1.03545−0.192185i1.03545 - 0.192185i
L(12)L(\frac12) ≈\approx 1.03545−0.192185i1.03545 - 0.192185i
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)
bad3 1+(1.41−0.998i)T 1 + (1.41 - 0.998i)T
5 1+(−2.05+0.891i)T 1 + (-2.05 + 0.891i)T
good2 1+(−0.199−0.0424i)T+(1.82+0.813i)T2 1 + (-0.199 - 0.0424i)T + (1.82 + 0.813i)T^{2}
7 1+(−0.530+0.918i)T+(−3.5−6.06i)T2 1 + (-0.530 + 0.918i)T + (-3.5 - 6.06i)T^{2}
11 1+(−5.73−1.21i)T+(10.0+4.47i)T2 1 + (-5.73 - 1.21i)T + (10.0 + 4.47i)T^{2}
13 1+(−3.20+0.680i)T+(11.8−5.28i)T2 1 + (-3.20 + 0.680i)T + (11.8 - 5.28i)T^{2}
17 1+(5.93+4.31i)T+(5.25+16.1i)T2 1 + (5.93 + 4.31i)T + (5.25 + 16.1i)T^{2}
19 1+(−2.26−1.64i)T+(5.87+18.0i)T2 1 + (-2.26 - 1.64i)T + (5.87 + 18.0i)T^{2}
23 1+(0.406−0.450i)T+(−2.40−22.8i)T2 1 + (0.406 - 0.450i)T + (-2.40 - 22.8i)T^{2}
29 1+(0.865+8.23i)T+(−28.3+6.02i)T2 1 + (0.865 + 8.23i)T + (-28.3 + 6.02i)T^{2}
31 1+(−0.439+4.18i)T+(−30.3−6.44i)T2 1 + (-0.439 + 4.18i)T + (-30.3 - 6.44i)T^{2}
37 1+(3.21−9.88i)T+(−29.9−21.7i)T2 1 + (3.21 - 9.88i)T + (-29.9 - 21.7i)T^{2}
41 1+(6.31−1.34i)T+(37.4−16.6i)T2 1 + (6.31 - 1.34i)T + (37.4 - 16.6i)T^{2}
43 1+(−1.32+2.29i)T+(−21.5−37.2i)T2 1 + (-1.32 + 2.29i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.0694−0.660i)T+(−45.9+9.77i)T2 1 + (-0.0694 - 0.660i)T + (-45.9 + 9.77i)T^{2}
53 1+(5.03−3.65i)T+(16.3−50.4i)T2 1 + (5.03 - 3.65i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.348+0.0741i)T+(53.8−23.9i)T2 1 + (-0.348 + 0.0741i)T + (53.8 - 23.9i)T^{2}
61 1+(−6.89−1.46i)T+(55.7+24.8i)T2 1 + (-6.89 - 1.46i)T + (55.7 + 24.8i)T^{2}
67 1+(1.16−11.0i)T+(−65.5−13.9i)T2 1 + (1.16 - 11.0i)T + (-65.5 - 13.9i)T^{2}
71 1+(3.31−2.40i)T+(21.9−67.5i)T2 1 + (3.31 - 2.40i)T + (21.9 - 67.5i)T^{2}
73 1+(2.32+7.14i)T+(−59.0+42.9i)T2 1 + (2.32 + 7.14i)T + (-59.0 + 42.9i)T^{2}
79 1+(−1.19−11.3i)T+(−77.2+16.4i)T2 1 + (-1.19 - 11.3i)T + (-77.2 + 16.4i)T^{2}
83 1+(−5.25+2.33i)T+(55.5−61.6i)T2 1 + (-5.25 + 2.33i)T + (55.5 - 61.6i)T^{2}
89 1+(−1.63−5.04i)T+(−72.0+52.3i)T2 1 + (-1.63 - 5.04i)T + (-72.0 + 52.3i)T^{2}
97 1+(0.0258+0.245i)T+(−94.8+20.1i)T2 1 + (0.0258 + 0.245i)T + (-94.8 + 20.1i)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

−12.07421288013796355204356186567, −11.25247690356011003259446749132, −10.01765175357027091164723209685, −9.464914984893710365086454288646, −8.689645605416081263959605222675, −6.65492383807812902070562642949, −5.90718968225677281331413206804, −4.74812401257303935191640970287, −3.99955750102440715885780761272, −1.17042111699141903886877279137, 1.64042683439239307954911923750, 3.72160717456142404466326548263, 5.10926966261467730272367024032, 6.18737831528035822995317230309, 6.91647678038246073238055105305, 8.644539389579292454496463711336, 9.130541393668006783736090193280, 10.59594575682720633428241134936, 11.40124971762307871509961137115, 12.39805383425495749833660870557

Graph of the ZZ-function along the critical line