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Properties

Label 2-704-11.3-c1-0-11
Degree 22
Conductor 704704
Sign 0.927−0.374i0.927 - 0.374i
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  + (0.118 + 0.363i)3-s + (2.61 + 1.90i)5-s + (0.618 − 1.90i)7-s + (2.30 − 1.67i)9-s + (0.309 + 3.30i)11-s + (1 − 0.726i)13-s + (−0.381 + 1.17i)15-s + (0.5 + 0.363i)17-s + (−1.80 − 5.56i)19-s + 0.763·21-s − 1.23·23-s + (1.69 + 5.20i)25-s + (1.80 + 1.31i)27-s + (−1.38 + 4.25i)29-s + (1.61 − 1.17i)31-s + ⋯
L(s)  = 1  + (0.0681 + 0.209i)3-s + (1.17 + 0.850i)5-s + (0.233 − 0.718i)7-s + (0.769 − 0.559i)9-s + (0.0931 + 0.995i)11-s + (0.277 − 0.201i)13-s + (−0.0986 + 0.303i)15-s + (0.121 + 0.0881i)17-s + (−0.415 − 1.27i)19-s + 0.166·21-s − 0.257·23-s + (0.338 + 1.04i)25-s + (0.348 + 0.252i)27-s + (−0.256 + 0.789i)29-s + (0.290 − 0.211i)31-s + ⋯

Functional equation

Λ(s)=(704s/2 ΓC(s) L(s)=((0.927−0.374i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(704s/2 ΓC(s+1/2) L(s)=((0.927−0.374i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 704704    =    26⋅112^{6} \cdot 11
Sign: 0.927−0.374i0.927 - 0.374i
Analytic conductor: 5.621465.62146
Root analytic conductor: 2.370962.37096
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ704(641,⋅)\chi_{704} (641, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 704, ( :1/2), 0.927−0.374i)(2,\ 704,\ (\ :1/2),\ 0.927 - 0.374i)

Particular Values

L(1)L(1) ≈\approx 2.01273+0.390943i2.01273 + 0.390943i
L(12)L(\frac12) ≈\approx 2.01273+0.390943i2.01273 + 0.390943i
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+(−0.309−3.30i)T 1 + (-0.309 - 3.30i)T
good3 1+(−0.118−0.363i)T+(−2.42+1.76i)T2 1 + (-0.118 - 0.363i)T + (-2.42 + 1.76i)T^{2}
5 1+(−2.61−1.90i)T+(1.54+4.75i)T2 1 + (-2.61 - 1.90i)T + (1.54 + 4.75i)T^{2}
7 1+(−0.618+1.90i)T+(−5.66−4.11i)T2 1 + (-0.618 + 1.90i)T + (-5.66 - 4.11i)T^{2}
13 1+(−1+0.726i)T+(4.01−12.3i)T2 1 + (-1 + 0.726i)T + (4.01 - 12.3i)T^{2}
17 1+(−0.5−0.363i)T+(5.25+16.1i)T2 1 + (-0.5 - 0.363i)T + (5.25 + 16.1i)T^{2}
19 1+(1.80+5.56i)T+(−15.3+11.1i)T2 1 + (1.80 + 5.56i)T + (-15.3 + 11.1i)T^{2}
23 1+1.23T+23T2 1 + 1.23T + 23T^{2}
29 1+(1.38−4.25i)T+(−23.4−17.0i)T2 1 + (1.38 - 4.25i)T + (-23.4 - 17.0i)T^{2}
31 1+(−1.61+1.17i)T+(9.57−29.4i)T2 1 + (-1.61 + 1.17i)T + (9.57 - 29.4i)T^{2}
37 1+(−1.14+3.52i)T+(−29.9−21.7i)T2 1 + (-1.14 + 3.52i)T + (-29.9 - 21.7i)T^{2}
41 1+(−1.73−5.34i)T+(−33.1+24.0i)T2 1 + (-1.73 - 5.34i)T + (-33.1 + 24.0i)T^{2}
43 1+8.56T+43T2 1 + 8.56T + 43T^{2}
47 1+(−2−6.15i)T+(−38.0+27.6i)T2 1 + (-2 - 6.15i)T + (-38.0 + 27.6i)T^{2}
53 1+(1.23−0.898i)T+(16.3−50.4i)T2 1 + (1.23 - 0.898i)T + (16.3 - 50.4i)T^{2}
59 1+(2.66−8.19i)T+(−47.7−34.6i)T2 1 + (2.66 - 8.19i)T + (-47.7 - 34.6i)T^{2}
61 1+(2+1.45i)T+(18.8+58.0i)T2 1 + (2 + 1.45i)T + (18.8 + 58.0i)T^{2}
67 1−11.0T+67T2 1 - 11.0T + 67T^{2}
71 1+(4.23+3.07i)T+(21.9+67.5i)T2 1 + (4.23 + 3.07i)T + (21.9 + 67.5i)T^{2}
73 1+(−3.20+9.87i)T+(−59.0−42.9i)T2 1 + (-3.20 + 9.87i)T + (-59.0 - 42.9i)T^{2}
79 1+(−10.8+7.88i)T+(24.4−75.1i)T2 1 + (-10.8 + 7.88i)T + (24.4 - 75.1i)T^{2}
83 1+(7.54+5.48i)T+(25.6+78.9i)T2 1 + (7.54 + 5.48i)T + (25.6 + 78.9i)T^{2}
89 1+8.09T+89T2 1 + 8.09T + 89T^{2}
97 1+(5.78−4.20i)T+(29.9−92.2i)T2 1 + (5.78 - 4.20i)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

−10.45796263000477255791782624319, −9.709126456513467266355842838340, −9.119812211891116903703166691643, −7.66782436890770610737793173452, −6.86593511442014058207956331485, −6.28581107934889659182686803034, −4.96375545083315788195811752620, −4.02059519121880717507133596323, −2.73149174884401650336538326864, −1.48894032084849305696760035562, 1.39336544754811113656699097141, 2.28737442491466018081406974505, 3.92845744287755153731773295868, 5.19422661765513422226681008222, 5.77377233876305941554149123651, 6.68332659587717983714232691594, 8.145976137612683929787730298673, 8.531683897154999850758051211526, 9.595217841023178047537425727696, 10.17087732892897904865455711655

Graph of the ZZ-function along the critical line