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Properties

Label 2-980-28.27-c1-0-1
Degree 22
Conductor 980980
Sign −0.0110+0.999i-0.0110 + 0.999i
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  + (0.486 + 1.32i)2-s − 0.812·3-s + (−1.52 + 1.29i)4-s + i·5-s + (−0.395 − 1.07i)6-s + (−2.45 − 1.39i)8-s − 2.34·9-s + (−1.32 + 0.486i)10-s + 4.86i·11-s + (1.23 − 1.04i)12-s − 0.895i·13-s − 0.812i·15-s + (0.658 − 3.94i)16-s − 5.89i·17-s + (−1.13 − 3.10i)18-s − 2.91·19-s + ⋯
L(s)  = 1  + (0.344 + 0.938i)2-s − 0.468·3-s + (−0.763 + 0.646i)4-s + 0.447i·5-s + (−0.161 − 0.440i)6-s + (−0.869 − 0.494i)8-s − 0.780·9-s + (−0.419 + 0.153i)10-s + 1.46i·11-s + (0.357 − 0.302i)12-s − 0.248i·13-s − 0.209i·15-s + (0.164 − 0.986i)16-s − 1.42i·17-s + (−0.268 − 0.732i)18-s − 0.669·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.0110+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0110 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.0110+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0110 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.0110+0.999i-0.0110 + 0.999i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(391,⋅)\chi_{980} (391, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.0110+0.999i)(2,\ 980,\ (\ :1/2),\ -0.0110 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.0708303−0.0716161i0.0708303 - 0.0716161i
L(12)L(\frac12) ≈\approx 0.0708303−0.0716161i0.0708303 - 0.0716161i
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+(−0.486−1.32i)T 1 + (-0.486 - 1.32i)T
5 1−iT 1 - iT
7 1 1
good3 1+0.812T+3T2 1 + 0.812T + 3T^{2}
11 1−4.86iT−11T2 1 - 4.86iT - 11T^{2}
13 1+0.895iT−13T2 1 + 0.895iT - 13T^{2}
17 1+5.89iT−17T2 1 + 5.89iT - 17T^{2}
19 1+2.91T+19T2 1 + 2.91T + 19T^{2}
23 1−1.56iT−23T2 1 - 1.56iT - 23T^{2}
29 1+9.73T+29T2 1 + 9.73T + 29T^{2}
31 1−4.41T+31T2 1 - 4.41T + 31T^{2}
37 1+1.82T+37T2 1 + 1.82T + 37T^{2}
41 1+10.4iT−41T2 1 + 10.4iT - 41T^{2}
43 1+3.04iT−43T2 1 + 3.04iT - 43T^{2}
47 1+5.21T+47T2 1 + 5.21T + 47T^{2}
53 1−0.179T+53T2 1 - 0.179T + 53T^{2}
59 1+11.3T+59T2 1 + 11.3T + 59T^{2}
61 1+6.15iT−61T2 1 + 6.15iT - 61T^{2}
67 1−7.79iT−67T2 1 - 7.79iT - 67T^{2}
71 1+8.38iT−71T2 1 + 8.38iT - 71T^{2}
73 1−8.78iT−73T2 1 - 8.78iT - 73T^{2}
79 1+3.63iT−79T2 1 + 3.63iT - 79T^{2}
83 1−2.03T+83T2 1 - 2.03T + 83T^{2}
89 1−1.76iT−89T2 1 - 1.76iT - 89T^{2}
97 1−7.83iT−97T2 1 - 7.83iT - 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

−10.58471526023154811900696924218, −9.603471998370721119221547457816, −8.950917969718390866300426412741, −7.79348240674655613535863612952, −7.17798461985891934502947651254, −6.42172407338993400124422838130, −5.44281358937629575420176394081, −4.80889878654573018611653044913, −3.65731781569277731383444570397, −2.43135570976499798493576260411, 0.04360922744047861354344297945, 1.50942821092903267038002189601, 2.92774194773503904404176407926, 3.88145238182561010607809894552, 4.89282266744385152498137212138, 5.93455383928205352334314206388, 6.22324454970133621593438104233, 8.149666267574933546456038521942, 8.602147632217234237133894604947, 9.458454745378026668509883547113

Graph of the ZZ-function along the critical line