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Properties

Label 2-980-35.9-c1-0-19
Degree 22
Conductor 980980
Sign −0.185+0.982i-0.185 + 0.982i
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)=((−0.185+0.982i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.185+0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.185+0.982i-0.185 + 0.982i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(569,⋅)\chi_{980} (569, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.185+0.982i)(2,\ 980,\ (\ :1/2),\ -0.185 + 0.982i)

Particular Values

L(1)L(1) ≈\approx 1.64746−1.98781i1.64746 - 1.98781i
L(12)L(\frac12) ≈\approx 1.64746−1.98781i1.64746 - 1.98781i
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

−9.551512034290190293324522526241, −8.667312171599813521900788112716, −8.158971344422432364903214444130, −7.66287278232485160802656445904, −6.45492950259355283937012833394, −5.51666822624580482865690215764, −4.18414552022219464378180601435, −3.25280095826607264231692118461, −2.13953807299700065005759460834, −1.05431700983908253886378584712, 2.21350996757624950219245770409, 2.84909332267963434753625479172, 3.84026228913242950129517929817, 4.65023441726917074162145194064, 5.94043774516656480927488681283, 7.21250933477752441130218541781, 7.75074184773112457121461601321, 8.657490878952809464549848455675, 9.445539821258872373899085040339, 10.26026453260722197668040557818

Graph of the ZZ-function along the critical line