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Properties

Label 2-980-35.4-c1-0-3
Degree 22
Conductor 980980
Sign 0.946−0.324i0.946 - 0.324i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
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  + (−2.59 − 1.5i)3-s + (−0.133 − 2.23i)5-s + (3 + 5.19i)9-s + (−1.5 + 2.59i)11-s + i·13-s + (−3 + 6i)15-s + (−4.33 − 2.5i)17-s + (4 + 6.92i)19-s + (−1.73 + i)23-s + (−4.96 + 0.598i)25-s − 9i·27-s + 29-s + (−1 + 1.73i)31-s + (7.79 − 4.5i)33-s + (8.66 − 5i)37-s + ⋯
L(s)  = 1  + (−1.49 − 0.866i)3-s + (−0.0599 − 0.998i)5-s + (1 + 1.73i)9-s + (−0.452 + 0.783i)11-s + 0.277i·13-s + (−0.774 + 1.54i)15-s + (−1.05 − 0.606i)17-s + (0.917 + 1.58i)19-s + (−0.361 + 0.208i)23-s + (−0.992 + 0.119i)25-s − 1.73i·27-s + 0.185·29-s + (−0.179 + 0.311i)31-s + (1.35 − 0.783i)33-s + (1.42 − 0.821i)37-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.946−0.324i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.946−0.324i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.946−0.324i0.946 - 0.324i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(949,⋅)\chi_{980} (949, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.946−0.324i)(2,\ 980,\ (\ :1/2),\ 0.946 - 0.324i)

Particular Values

L(1)L(1) ≈\approx 0.604357+0.100649i0.604357 + 0.100649i
L(12)L(\frac12) ≈\approx 0.604357+0.100649i0.604357 + 0.100649i
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
5 1+(0.133+2.23i)T 1 + (0.133 + 2.23i)T
7 1 1
good3 1+(2.59+1.5i)T+(1.5+2.59i)T2 1 + (2.59 + 1.5i)T + (1.5 + 2.59i)T^{2}
11 1+(1.5−2.59i)T+(−5.5−9.52i)T2 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2}
13 1−iT−13T2 1 - iT - 13T^{2}
17 1+(4.33+2.5i)T+(8.5+14.7i)T2 1 + (4.33 + 2.5i)T + (8.5 + 14.7i)T^{2}
19 1+(−4−6.92i)T+(−9.5+16.4i)T2 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.73−i)T+(11.5−19.9i)T2 1 + (1.73 - i)T + (11.5 - 19.9i)T^{2}
29 1−T+29T2 1 - T + 29T^{2}
31 1+(1−1.73i)T+(−15.5−26.8i)T2 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2}
37 1+(−8.66+5i)T+(18.5−32.0i)T2 1 + (-8.66 + 5i)T + (18.5 - 32.0i)T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1+(9.52−5.5i)T+(23.5−40.7i)T2 1 + (9.52 - 5.5i)T + (23.5 - 40.7i)T^{2}
53 1+(−5.19−3i)T+(26.5+45.8i)T2 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2}
59 1+(−5+8.66i)T+(−29.5−51.0i)T2 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2}
61 1+(−30.5+52.8i)T2 1 + (-30.5 + 52.8i)T^{2}
67 1+(−8.66−5i)T+(33.5+58.0i)T2 1 + (-8.66 - 5i)T + (33.5 + 58.0i)T^{2}
71 1+71T2 1 + 71T^{2}
73 1+(−8.66−5i)T+(36.5+63.2i)T2 1 + (-8.66 - 5i)T + (36.5 + 63.2i)T^{2}
79 1+(3.5+6.06i)T+(−39.5+68.4i)T2 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1+(4+6.92i)T+(−44.5+77.0i)T2 1 + (4 + 6.92i)T + (-44.5 + 77.0i)T^{2}
97 1+3iT−97T2 1 + 3iT - 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

−10.04984564198975422885161118723, −9.390437329431995130681000048556, −8.060908302065309439457015467889, −7.50332231487858494944087204237, −6.54246563685918802109861910950, −5.69432002156047575343929288719, −5.02270565418853954589469780708, −4.17458753199840679756951925377, −2.10716521083837070371705689821, −1.02484058119904222543029692529, 0.42636961076067710366381340074, 2.67915935985273201602604406358, 3.82845209086311144048413634782, 4.78820270560852647865282149702, 5.67800944691663403298356849737, 6.38811216024287723597101720380, 7.10365631757318600519899018152, 8.317371218635312220296071027912, 9.489162150510769453158272970996, 10.15665497531042525440201299405

Graph of the ZZ-function along the critical line