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Properties

Label 2-980-7.2-c3-0-35
Degree 22
Conductor 980980
Sign −0.701−0.712i-0.701 - 0.712i
Analytic cond. 57.821857.8218
Root an. cond. 7.604067.60406
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.5 − 4.33i)3-s + (2.5 − 4.33i)5-s + (0.999 − 1.73i)9-s + (7.5 + 12.9i)11-s + 13·13-s − 25.0·15-s + (−13.5 − 23.3i)17-s + (−77 + 133. i)19-s + (93 − 161. i)23-s + (−12.5 − 21.6i)25-s − 144.·27-s + 3·29-s + (−164 − 284. i)31-s + (37.5 − 64.9i)33-s + (−127 + 219. i)37-s + ⋯
L(s)  = 1  + (−0.481 − 0.833i)3-s + (0.223 − 0.387i)5-s + (0.0370 − 0.0641i)9-s + (0.205 + 0.356i)11-s + 0.277·13-s − 0.430·15-s + (−0.192 − 0.333i)17-s + (−0.929 + 1.61i)19-s + (0.843 − 1.46i)23-s + (−0.100 − 0.173i)25-s − 1.03·27-s + 0.0192·29-s + (−0.950 − 1.64i)31-s + (0.197 − 0.342i)33-s + (−0.564 + 0.977i)37-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.701−0.712i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+3/2) L(s)=((−0.701−0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.701−0.712i-0.701 - 0.712i
Analytic conductor: 57.821857.8218
Root analytic conductor: 7.604067.60406
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ980(961,⋅)\chi_{980} (961, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :3/2), −0.701−0.712i)(2,\ 980,\ (\ :3/2),\ -0.701 - 0.712i)

Particular Values

L(2)L(2) ≈\approx 0.32675822270.3267582227
L(12)L(\frac12) ≈\approx 0.32675822270.3267582227
L(52)L(\frac{5}{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+(−2.5+4.33i)T 1 + (-2.5 + 4.33i)T
7 1 1
good3 1+(2.5+4.33i)T+(−13.5+23.3i)T2 1 + (2.5 + 4.33i)T + (-13.5 + 23.3i)T^{2}
11 1+(−7.5−12.9i)T+(−665.5+1.15e3i)T2 1 + (-7.5 - 12.9i)T + (-665.5 + 1.15e3i)T^{2}
13 1−13T+2.19e3T2 1 - 13T + 2.19e3T^{2}
17 1+(13.5+23.3i)T+(−2.45e3+4.25e3i)T2 1 + (13.5 + 23.3i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(77−133.i)T+(−3.42e3−5.94e3i)T2 1 + (77 - 133. i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(−93+161.i)T+(−6.08e3−1.05e4i)T2 1 + (-93 + 161. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1−3T+2.43e4T2 1 - 3T + 2.43e4T^{2}
31 1+(164+284.i)T+(−1.48e4+2.57e4i)T2 1 + (164 + 284. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(127−219.i)T+(−2.53e4−4.38e4i)T2 1 + (127 - 219. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1+96T+6.89e4T2 1 + 96T + 6.89e4T^{2}
43 1−134T+7.95e4T2 1 - 134T + 7.95e4T^{2}
47 1+(−25.5+44.1i)T+(−5.19e4−8.99e4i)T2 1 + (-25.5 + 44.1i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(120+207.i)T+(−7.44e4+1.28e5i)T2 1 + (120 + 207. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(198+342.i)T+(−1.02e5+1.77e5i)T2 1 + (198 + 342. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(308−533.i)T+(−1.13e5−1.96e5i)T2 1 + (308 - 533. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(148+256.i)T+(−1.50e5+2.60e5i)T2 1 + (148 + 256. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+48T+3.57e5T2 1 + 48T + 3.57e5T^{2}
73 1+(161+278.i)T+(−1.94e5+3.36e5i)T2 1 + (161 + 278. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(329.5−570.i)T+(−2.46e5−4.26e5i)T2 1 + (329.5 - 570. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+300T+5.71e5T2 1 + 300T + 5.71e5T^{2}
89 1+(−510+883.i)T+(−3.52e5−6.10e5i)T2 1 + (-510 + 883. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1−199T+9.12e5T2 1 - 199T + 9.12e5T^{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

−9.045421544376108222864052123903, −8.207718243924617072248016641471, −7.31672064278767358603216596795, −6.43141615348541834869962123388, −5.90103358176555698201045829837, −4.72546423493906028613694773667, −3.75431605309231219861817224812, −2.20627018580590331258230655295, −1.27327542420351744444946177677, −0.088561251189226392705214232137, 1.61232492289303765167476126310, 2.99959866714197180010797992242, 3.99927793883202316040319704788, 4.96496748883927920799702953109, 5.69003541410506476139286667585, 6.72995060631504894236144495922, 7.47258036429880613369264215433, 8.815583880703080147166424819927, 9.255109555070225102644864004464, 10.32323225063286106676204367390

Graph of the ZZ-function along the critical line