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Properties

Label 2-264-11.3-c1-0-3
Degree 22
Conductor 264264
Sign 0.751+0.659i0.751 + 0.659i
Analytic cond. 2.108052.10805
Root an. cond. 1.451911.45191
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  + (−0.309 − 0.951i)3-s + (1.30 + 0.951i)5-s + (0.690 − 2.12i)7-s + (−0.809 + 0.587i)9-s + (2.80 − 1.76i)11-s + (0.190 − 0.138i)13-s + (0.499 − 1.53i)15-s + (2.11 + 1.53i)17-s + (−1.11 − 3.44i)19-s − 2.23·21-s + 3.47·23-s + (−0.736 − 2.26i)25-s + (0.809 + 0.587i)27-s + (0.618 − 1.90i)29-s + (−2.5 + 1.81i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (0.585 + 0.425i)5-s + (0.261 − 0.803i)7-s + (−0.269 + 0.195i)9-s + (0.846 − 0.531i)11-s + (0.0529 − 0.0384i)13-s + (0.129 − 0.397i)15-s + (0.513 + 0.373i)17-s + (−0.256 − 0.789i)19-s − 0.487·21-s + 0.723·23-s + (−0.147 − 0.453i)25-s + (0.155 + 0.113i)27-s + (0.114 − 0.353i)29-s + (−0.449 + 0.326i)31-s + ⋯

Functional equation

Λ(s)=(264s/2 ΓC(s) L(s)=((0.751+0.659i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(264s/2 ΓC(s+1/2) L(s)=((0.751+0.659i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 264264    =    23⋅3⋅112^{3} \cdot 3 \cdot 11
Sign: 0.751+0.659i0.751 + 0.659i
Analytic conductor: 2.108052.10805
Root analytic conductor: 1.451911.45191
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ264(25,⋅)\chi_{264} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 264, ( :1/2), 0.751+0.659i)(2,\ 264,\ (\ :1/2),\ 0.751 + 0.659i)

Particular Values

L(1)L(1) ≈\approx 1.27461−0.479650i1.27461 - 0.479650i
L(12)L(\frac12) ≈\approx 1.27461−0.479650i1.27461 - 0.479650i
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+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
11 1+(−2.80+1.76i)T 1 + (-2.80 + 1.76i)T
good5 1+(−1.30−0.951i)T+(1.54+4.75i)T2 1 + (-1.30 - 0.951i)T + (1.54 + 4.75i)T^{2}
7 1+(−0.690+2.12i)T+(−5.66−4.11i)T2 1 + (-0.690 + 2.12i)T + (-5.66 - 4.11i)T^{2}
13 1+(−0.190+0.138i)T+(4.01−12.3i)T2 1 + (-0.190 + 0.138i)T + (4.01 - 12.3i)T^{2}
17 1+(−2.11−1.53i)T+(5.25+16.1i)T2 1 + (-2.11 - 1.53i)T + (5.25 + 16.1i)T^{2}
19 1+(1.11+3.44i)T+(−15.3+11.1i)T2 1 + (1.11 + 3.44i)T + (-15.3 + 11.1i)T^{2}
23 1−3.47T+23T2 1 - 3.47T + 23T^{2}
29 1+(−0.618+1.90i)T+(−23.4−17.0i)T2 1 + (-0.618 + 1.90i)T + (-23.4 - 17.0i)T^{2}
31 1+(2.5−1.81i)T+(9.57−29.4i)T2 1 + (2.5 - 1.81i)T + (9.57 - 29.4i)T^{2}
37 1+(2.07−6.37i)T+(−29.9−21.7i)T2 1 + (2.07 - 6.37i)T + (-29.9 - 21.7i)T^{2}
41 1+(−2.54−7.83i)T+(−33.1+24.0i)T2 1 + (-2.54 - 7.83i)T + (-33.1 + 24.0i)T^{2}
43 1+11.9T+43T2 1 + 11.9T + 43T^{2}
47 1+(−2.28−7.02i)T+(−38.0+27.6i)T2 1 + (-2.28 - 7.02i)T + (-38.0 + 27.6i)T^{2}
53 1+(9.35−6.79i)T+(16.3−50.4i)T2 1 + (9.35 - 6.79i)T + (16.3 - 50.4i)T^{2}
59 1+(0.881−2.71i)T+(−47.7−34.6i)T2 1 + (0.881 - 2.71i)T + (-47.7 - 34.6i)T^{2}
61 1+(−3.54−2.57i)T+(18.8+58.0i)T2 1 + (-3.54 - 2.57i)T + (18.8 + 58.0i)T^{2}
67 1+14.0T+67T2 1 + 14.0T + 67T^{2}
71 1+(−7.16−5.20i)T+(21.9+67.5i)T2 1 + (-7.16 - 5.20i)T + (21.9 + 67.5i)T^{2}
73 1+(−4.09+12.5i)T+(−59.0−42.9i)T2 1 + (-4.09 + 12.5i)T + (-59.0 - 42.9i)T^{2}
79 1+(−1.42+1.03i)T+(24.4−75.1i)T2 1 + (-1.42 + 1.03i)T + (24.4 - 75.1i)T^{2}
83 1+(7.42+5.39i)T+(25.6+78.9i)T2 1 + (7.42 + 5.39i)T + (25.6 + 78.9i)T^{2}
89 1+12.2T+89T2 1 + 12.2T + 89T^{2}
97 1+(−4.78+3.47i)T+(29.9−92.2i)T2 1 + (-4.78 + 3.47i)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

−11.75129053428547926876570532768, −10.95476002851036419846964367513, −10.09837448328689119638345741761, −8.945014666116103270334596193109, −7.83616669167496805942455157985, −6.76758275980106952995241482690, −6.06352948763283613123486280426, −4.60433315582000019413930866466, −3.08501240962548781242511157968, −1.34163356364379587096684064627, 1.85610088045989469098779786115, 3.64198918007368013772081048300, 5.03673494993019992400509103078, 5.76183152306043092707620958607, 7.04150843403177993492083843688, 8.504788972906800848878555437579, 9.272766915912266130439186058178, 9.995263775747147677570193985854, 11.17876726681107133467949827325, 12.06860877175244176979950851600

Graph of the ZZ-function along the critical line