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Properties

Label 2-72-72.13-c1-0-2
Degree 22
Conductor 7272
Sign 0.791−0.610i0.791 - 0.610i
Analytic cond. 0.5749220.574922
Root an. cond. 0.7582360.758236
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  + (−1.34 + 0.436i)2-s + (1.52 + 0.816i)3-s + (1.61 − 1.17i)4-s + (−0.602 − 0.348i)5-s + (−2.41 − 0.431i)6-s + (0.795 + 1.37i)7-s + (−1.66 + 2.28i)8-s + (1.66 + 2.49i)9-s + (0.962 + 0.205i)10-s + (2.37 − 1.36i)11-s + (3.43 − 0.470i)12-s + (−4.76 − 2.75i)13-s + (−1.67 − 1.50i)14-s + (−0.636 − 1.02i)15-s + (1.24 − 3.80i)16-s − 5.65·17-s + ⋯
L(s)  = 1  + (−0.951 + 0.308i)2-s + (0.882 + 0.471i)3-s + (0.809 − 0.586i)4-s + (−0.269 − 0.155i)5-s + (−0.984 − 0.176i)6-s + (0.300 + 0.520i)7-s + (−0.589 + 0.807i)8-s + (0.555 + 0.831i)9-s + (0.304 + 0.0649i)10-s + (0.715 − 0.412i)11-s + (0.990 − 0.135i)12-s + (−1.32 − 0.763i)13-s + (−0.446 − 0.402i)14-s + (−0.164 − 0.264i)15-s + (0.311 − 0.950i)16-s − 1.37·17-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((0.791−0.610i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+1/2) L(s)=((0.791−0.610i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 0.791−0.610i0.791 - 0.610i
Analytic conductor: 0.5749220.574922
Root analytic conductor: 0.7582360.758236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ72(13,⋅)\chi_{72} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :1/2), 0.791−0.610i)(2,\ 72,\ (\ :1/2),\ 0.791 - 0.610i)

Particular Values

L(1)L(1) ≈\approx 0.742676+0.253136i0.742676 + 0.253136i
L(12)L(\frac12) ≈\approx 0.742676+0.253136i0.742676 + 0.253136i
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+(1.34−0.436i)T 1 + (1.34 - 0.436i)T
3 1+(−1.52−0.816i)T 1 + (-1.52 - 0.816i)T
good5 1+(0.602+0.348i)T+(2.5+4.33i)T2 1 + (0.602 + 0.348i)T + (2.5 + 4.33i)T^{2}
7 1+(−0.795−1.37i)T+(−3.5+6.06i)T2 1 + (-0.795 - 1.37i)T + (-3.5 + 6.06i)T^{2}
11 1+(−2.37+1.36i)T+(5.5−9.52i)T2 1 + (-2.37 + 1.36i)T + (5.5 - 9.52i)T^{2}
13 1+(4.76+2.75i)T+(6.5+11.2i)T2 1 + (4.76 + 2.75i)T + (6.5 + 11.2i)T^{2}
17 1+5.65T+17T2 1 + 5.65T + 17T^{2}
19 1+0.963iT−19T2 1 + 0.963iT - 19T^{2}
23 1+(−3.28+5.69i)T+(−11.5−19.9i)T2 1 + (-3.28 + 5.69i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.85+1.64i)T+(14.5−25.1i)T2 1 + (-2.85 + 1.64i)T + (14.5 - 25.1i)T^{2}
31 1+(3.69−6.40i)T+(−15.5−26.8i)T2 1 + (3.69 - 6.40i)T + (-15.5 - 26.8i)T^{2}
37 1−6.25iT−37T2 1 - 6.25iT - 37T^{2}
41 1+(0.931−1.61i)T+(−20.5−35.5i)T2 1 + (0.931 - 1.61i)T + (-20.5 - 35.5i)T^{2}
43 1+(2.99−1.73i)T+(21.5−37.2i)T2 1 + (2.99 - 1.73i)T + (21.5 - 37.2i)T^{2}
47 1+(−3.85−6.67i)T+(−23.5+40.7i)T2 1 + (-3.85 - 6.67i)T + (-23.5 + 40.7i)T^{2}
53 1−2.54iT−53T2 1 - 2.54iT - 53T^{2}
59 1+(4.62+2.66i)T+(29.5+51.0i)T2 1 + (4.62 + 2.66i)T + (29.5 + 51.0i)T^{2}
61 1+(−7.93+4.58i)T+(30.5−52.8i)T2 1 + (-7.93 + 4.58i)T + (30.5 - 52.8i)T^{2}
67 1+(−5.95−3.43i)T+(33.5+58.0i)T2 1 + (-5.95 - 3.43i)T + (33.5 + 58.0i)T^{2}
71 1−3.68T+71T2 1 - 3.68T + 71T^{2}
73 1−2.83T+73T2 1 - 2.83T + 73T^{2}
79 1+(−2.87−4.98i)T+(−39.5+68.4i)T2 1 + (-2.87 - 4.98i)T + (-39.5 + 68.4i)T^{2}
83 1+(5.74−3.31i)T+(41.5−71.8i)T2 1 + (5.74 - 3.31i)T + (41.5 - 71.8i)T^{2}
89 1+2.98T+89T2 1 + 2.98T + 89T^{2}
97 1+(1.24+2.16i)T+(−48.5+84.0i)T2 1 + (1.24 + 2.16i)T + (-48.5 + 84.0i)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

−15.01767830975755357782350099009, −14.13694678763154232469931892792, −12.46192828103001619322008529334, −11.13082072871743772722909188224, −9.992085852300930481645130156985, −8.880465908744216968842542154310, −8.193867575294066097225624339531, −6.79504590041498687806396072087, −4.85663134523370781838398484345, −2.57014684954820428635206915791, 2.04116080153799008088038296487, 3.93334731717163109013865403712, 6.95109993138513701164004883467, 7.48668813279105744966084957247, 8.948149587374605967982285121504, 9.692148190626114705223404215515, 11.21324480270749118581905264935, 12.17191399222121318615344336683, 13.39513652513261198998576543838, 14.66702673473994531238952445891

Graph of the ZZ-function along the critical line