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Properties

Label 2-468-117.94-c1-0-5
Degree 22
Conductor 468468
Sign −0.551−0.834i-0.551 - 0.834i
Analytic cond. 3.736993.73699
Root an. cond. 1.933131.93313
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  + (0.583 + 1.63i)3-s + (−1.31 + 2.27i)5-s + 3.59·7-s + (−2.31 + 1.90i)9-s + (−2.37 + 4.11i)11-s + (−3.08 − 1.87i)13-s + (−4.47 − 0.813i)15-s + (0.733 − 1.27i)17-s + (2.78 − 4.82i)19-s + (2.09 + 5.85i)21-s + 3.41·23-s + (−0.942 − 1.63i)25-s + (−4.45 − 2.66i)27-s + (−1.91 + 3.32i)29-s + (−4.73 + 8.20i)31-s + ⋯
L(s)  = 1  + (0.336 + 0.941i)3-s + (−0.586 + 1.01i)5-s + 1.35·7-s + (−0.772 + 0.634i)9-s + (−0.716 + 1.24i)11-s + (−0.854 − 0.519i)13-s + (−1.15 − 0.209i)15-s + (0.177 − 0.308i)17-s + (0.639 − 1.10i)19-s + (0.457 + 1.27i)21-s + 0.711·23-s + (−0.188 − 0.326i)25-s + (−0.857 − 0.513i)27-s + (−0.356 + 0.617i)29-s + (−0.850 + 1.47i)31-s + ⋯

Functional equation

Λ(s)=(468s/2 ΓC(s) L(s)=((−0.551−0.834i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(468s/2 ΓC(s+1/2) L(s)=((−0.551−0.834i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 468468    =    22⋅32⋅132^{2} \cdot 3^{2} \cdot 13
Sign: −0.551−0.834i-0.551 - 0.834i
Analytic conductor: 3.736993.73699
Root analytic conductor: 1.933131.93313
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ468(445,⋅)\chi_{468} (445, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 468, ( :1/2), −0.551−0.834i)(2,\ 468,\ (\ :1/2),\ -0.551 - 0.834i)

Particular Values

L(1)L(1) ≈\approx 0.658864+1.22487i0.658864 + 1.22487i
L(12)L(\frac12) ≈\approx 0.658864+1.22487i0.658864 + 1.22487i
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 1
3 1+(−0.583−1.63i)T 1 + (-0.583 - 1.63i)T
13 1+(3.08+1.87i)T 1 + (3.08 + 1.87i)T
good5 1+(1.31−2.27i)T+(−2.5−4.33i)T2 1 + (1.31 - 2.27i)T + (-2.5 - 4.33i)T^{2}
7 1−3.59T+7T2 1 - 3.59T + 7T^{2}
11 1+(2.37−4.11i)T+(−5.5−9.52i)T2 1 + (2.37 - 4.11i)T + (-5.5 - 9.52i)T^{2}
17 1+(−0.733+1.27i)T+(−8.5−14.7i)T2 1 + (-0.733 + 1.27i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.78+4.82i)T+(−9.5−16.4i)T2 1 + (-2.78 + 4.82i)T + (-9.5 - 16.4i)T^{2}
23 1−3.41T+23T2 1 - 3.41T + 23T^{2}
29 1+(1.91−3.32i)T+(−14.5−25.1i)T2 1 + (1.91 - 3.32i)T + (-14.5 - 25.1i)T^{2}
31 1+(4.73−8.20i)T+(−15.5−26.8i)T2 1 + (4.73 - 8.20i)T + (-15.5 - 26.8i)T^{2}
37 1+(−2.14−3.71i)T+(−18.5+32.0i)T2 1 + (-2.14 - 3.71i)T + (-18.5 + 32.0i)T^{2}
41 1−7.34T+41T2 1 - 7.34T + 41T^{2}
43 1+2.56T+43T2 1 + 2.56T + 43T^{2}
47 1+(3.88+6.72i)T+(−23.5+40.7i)T2 1 + (3.88 + 6.72i)T + (-23.5 + 40.7i)T^{2}
53 1−1.77T+53T2 1 - 1.77T + 53T^{2}
59 1+(−4.39−7.61i)T+(−29.5+51.0i)T2 1 + (-4.39 - 7.61i)T + (-29.5 + 51.0i)T^{2}
61 1−6.37T+61T2 1 - 6.37T + 61T^{2}
67 1−4.24T+67T2 1 - 4.24T + 67T^{2}
71 1+(−4.76+8.25i)T+(−35.5−61.4i)T2 1 + (-4.76 + 8.25i)T + (-35.5 - 61.4i)T^{2}
73 1−11.1T+73T2 1 - 11.1T + 73T^{2}
79 1+(5.06+8.77i)T+(−39.5+68.4i)T2 1 + (5.06 + 8.77i)T + (-39.5 + 68.4i)T^{2}
83 1+(4.51+7.82i)T+(−41.5+71.8i)T2 1 + (4.51 + 7.82i)T + (-41.5 + 71.8i)T^{2}
89 1+(−6.36−11.0i)T+(−44.5+77.0i)T2 1 + (-6.36 - 11.0i)T + (-44.5 + 77.0i)T^{2}
97 1−12.0T+97T2 1 - 12.0T + 97T^{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

−11.11256873685278709790483578110, −10.55815649132971929796167539879, −9.702741807337137379612308704994, −8.639298563046091521517835374814, −7.53449427390448780721486922209, −7.20533911809244551103790573617, −5.06979987162710466318783328128, −4.88455212214592269913145237576, −3.35905582766403615311797268895, −2.36534842579509694951653473628, 0.850038336254408669699253402061, 2.21500138421751393530955562503, 3.82189424461650206549215241703, 5.07496096719836766670401671952, 5.89218066117836590522277585419, 7.50386952929960349836789935562, 7.976662942170808719709059286513, 8.566342684199211061495013977796, 9.589660194112179952111182937792, 11.23393044115097097152678962891

Graph of the ZZ-function along the critical line