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Properties

Label 2-704-88.83-c1-0-5
Degree 22
Conductor 704704
Sign −0.674−0.738i-0.674 - 0.738i
Analytic cond. 5.621465.62146
Root an. cond. 2.370962.37096
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  + (2.11 + 1.53i)3-s + (−1.17 − 0.381i)5-s + (−3.07 + 2.23i)7-s + (1.19 + 3.66i)9-s + (−2.80 + 1.76i)11-s + (−1.90 − 2.61i)15-s + (3.35 + 1.08i)17-s + (−2.07 + 2.85i)19-s − 9.95·21-s + 6i·23-s + (−2.80 − 2.04i)25-s + (−0.690 + 2.12i)27-s + (−4.25 + 3.09i)29-s + (6.88 − 2.23i)31-s + (−8.66 − 0.587i)33-s + ⋯
L(s)  = 1  + (1.22 + 0.888i)3-s + (−0.525 − 0.170i)5-s + (−1.16 + 0.845i)7-s + (0.396 + 1.22i)9-s + (−0.846 + 0.531i)11-s + (−0.491 − 0.675i)15-s + (0.813 + 0.264i)17-s + (−0.475 + 0.654i)19-s − 2.17·21-s + 1.25i·23-s + (−0.561 − 0.408i)25-s + (−0.132 + 0.409i)27-s + (−0.789 + 0.573i)29-s + (1.23 − 0.401i)31-s + (−1.50 − 0.102i)33-s + ⋯

Functional equation

Λ(s)=(704s/2 ΓC(s) L(s)=((−0.674−0.738i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(704s/2 ΓC(s+1/2) L(s)=((−0.674−0.738i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 704704    =    26⋅112^{6} \cdot 11
Sign: −0.674−0.738i-0.674 - 0.738i
Analytic conductor: 5.621465.62146
Root analytic conductor: 2.370962.37096
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ704(479,⋅)\chi_{704} (479, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 704, ( :1/2), −0.674−0.738i)(2,\ 704,\ (\ :1/2),\ -0.674 - 0.738i)

Particular Values

L(1)L(1) ≈\approx 0.574477+1.30184i0.574477 + 1.30184i
L(12)L(\frac12) ≈\approx 0.574477+1.30184i0.574477 + 1.30184i
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
11 1+(2.80−1.76i)T 1 + (2.80 - 1.76i)T
good3 1+(−2.11−1.53i)T+(0.927+2.85i)T2 1 + (-2.11 - 1.53i)T + (0.927 + 2.85i)T^{2}
5 1+(1.17+0.381i)T+(4.04+2.93i)T2 1 + (1.17 + 0.381i)T + (4.04 + 2.93i)T^{2}
7 1+(3.07−2.23i)T+(2.16−6.65i)T2 1 + (3.07 - 2.23i)T + (2.16 - 6.65i)T^{2}
13 1+(−10.5+7.64i)T2 1 + (-10.5 + 7.64i)T^{2}
17 1+(−3.35−1.08i)T+(13.7+9.99i)T2 1 + (-3.35 - 1.08i)T + (13.7 + 9.99i)T^{2}
19 1+(2.07−2.85i)T+(−5.87−18.0i)T2 1 + (2.07 - 2.85i)T + (-5.87 - 18.0i)T^{2}
23 1−6iT−23T2 1 - 6iT - 23T^{2}
29 1+(4.25−3.09i)T+(8.96−27.5i)T2 1 + (4.25 - 3.09i)T + (8.96 - 27.5i)T^{2}
31 1+(−6.88+2.23i)T+(25.0−18.2i)T2 1 + (-6.88 + 2.23i)T + (25.0 - 18.2i)T^{2}
37 1+(−2.17−3i)T+(−11.4+35.1i)T2 1 + (-2.17 - 3i)T + (-11.4 + 35.1i)T^{2}
41 1+(−3.35+4.61i)T+(−12.6−38.9i)T2 1 + (-3.35 + 4.61i)T + (-12.6 - 38.9i)T^{2}
43 1−12.7iT−43T2 1 - 12.7iT - 43T^{2}
47 1+(−5.70+7.85i)T+(−14.5−44.6i)T2 1 + (-5.70 + 7.85i)T + (-14.5 - 44.6i)T^{2}
53 1+(3.07−i)T+(42.8−31.1i)T2 1 + (3.07 - i)T + (42.8 - 31.1i)T^{2}
59 1+(−8.39+6.10i)T+(18.2−56.1i)T2 1 + (-8.39 + 6.10i)T + (18.2 - 56.1i)T^{2}
61 1+(−3.52+10.8i)T+(−49.3−35.8i)T2 1 + (-3.52 + 10.8i)T + (-49.3 - 35.8i)T^{2}
67 1+14.5T+67T2 1 + 14.5T + 67T^{2}
71 1+(−11.4−3.70i)T+(57.4+41.7i)T2 1 + (-11.4 - 3.70i)T + (57.4 + 41.7i)T^{2}
73 1+(0.590+0.812i)T+(−22.5+69.4i)T2 1 + (0.590 + 0.812i)T + (-22.5 + 69.4i)T^{2}
79 1+(−1.45−4.47i)T+(−63.9+46.4i)T2 1 + (-1.45 - 4.47i)T + (-63.9 + 46.4i)T^{2}
83 1+(0.427+0.138i)T+(67.1+48.7i)T2 1 + (0.427 + 0.138i)T + (67.1 + 48.7i)T^{2}
89 1−1.85T+89T2 1 - 1.85T + 89T^{2}
97 1+(−2.57−7.91i)T+(−78.4+57.0i)T2 1 + (-2.57 - 7.91i)T + (-78.4 + 57.0i)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

−10.32045409149158447937619964151, −9.767262066533916556223228807950, −9.201355838601455622256566792377, −8.178059126667805983489726358365, −7.70838320909545344309751359477, −6.25717881535089602159353792631, −5.21269034509867009846261279524, −3.99317153873888929063525718285, −3.27503150296644360062005141052, −2.32173378986605508682491081513, 0.61783188027387094603572973085, 2.51854160385647139069969964040, 3.23004168097470182971668194535, 4.21612983338860037628170059177, 5.90052801333211004223245654161, 6.95581919951927654898409282197, 7.50781584046559185987019879526, 8.250806473399698859689067360079, 9.102709125953566633052105452565, 10.07347092403812978773815201524

Graph of the ZZ-function along the critical line