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Properties

Label 2-980-140.59-c1-0-38
Degree 22
Conductor 980980
Sign −0.997−0.0633i-0.997 - 0.0633i
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  + (−0.707 − 1.22i)2-s + (−2.73 − 1.58i)3-s + (−0.999 + 1.73i)4-s + (−1.93 + 1.11i)5-s + 4.47i·6-s + 2.82·8-s + (3.5 + 6.06i)9-s + (2.73 + 1.58i)10-s + (5.47 − 3.16i)12-s + 7.07·15-s + (−2.00 − 3.46i)16-s + (4.94 − 8.57i)18-s − 4.47i·20-s + (0.707 + 1.22i)23-s + (−7.74 − 4.47i)24-s + (2.5 − 4.33i)25-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (−1.58 − 0.912i)3-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.499i)5-s + 1.82i·6-s + 0.999·8-s + (1.16 + 2.02i)9-s + (0.866 + 0.499i)10-s + (1.58 − 0.912i)12-s + 1.82·15-s + (−0.500 − 0.866i)16-s + (1.16 − 2.02i)18-s − 0.999i·20-s + (0.147 + 0.255i)23-s + (−1.58 − 0.912i)24-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.997−0.0633i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.997−0.0633i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.997−0.0633i-0.997 - 0.0633i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(619,⋅)\chi_{980} (619, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.997−0.0633i)(2,\ 980,\ (\ :1/2),\ -0.997 - 0.0633i)

Particular Values

L(1)L(1) ≈\approx 0.00753872+0.237828i0.00753872 + 0.237828i
L(12)L(\frac12) ≈\approx 0.00753872+0.237828i0.00753872 + 0.237828i
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+(0.707+1.22i)T 1 + (0.707 + 1.22i)T
5 1+(1.93−1.11i)T 1 + (1.93 - 1.11i)T
7 1 1
good3 1+(2.73+1.58i)T+(1.5+2.59i)T2 1 + (2.73 + 1.58i)T + (1.5 + 2.59i)T^{2}
11 1+(5.5+9.52i)T2 1 + (5.5 + 9.52i)T^{2}
13 1+13T2 1 + 13T^{2}
17 1+(−8.5−14.7i)T2 1 + (-8.5 - 14.7i)T^{2}
19 1+(−9.5+16.4i)T2 1 + (-9.5 + 16.4i)T^{2}
23 1+(−0.707−1.22i)T+(−11.5+19.9i)T2 1 + (-0.707 - 1.22i)T + (-11.5 + 19.9i)T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+(−15.5−26.8i)T2 1 + (-15.5 - 26.8i)T^{2}
37 1+(18.5−32.0i)T2 1 + (18.5 - 32.0i)T^{2}
41 1−4.47iT−41T2 1 - 4.47iT - 41T^{2}
43 1+12.7T+43T2 1 + 12.7T + 43T^{2}
47 1+(−8.21+4.74i)T+(23.5−40.7i)T2 1 + (-8.21 + 4.74i)T + (23.5 - 40.7i)T^{2}
53 1+(26.5+45.8i)T2 1 + (26.5 + 45.8i)T^{2}
59 1+(−29.5−51.0i)T2 1 + (-29.5 - 51.0i)T^{2}
61 1+(11.6−6.70i)T+(30.5−52.8i)T2 1 + (11.6 - 6.70i)T + (30.5 - 52.8i)T^{2}
67 1+(2.12−3.67i)T+(−33.5−58.0i)T2 1 + (2.12 - 3.67i)T + (-33.5 - 58.0i)T^{2}
71 1−71T2 1 - 71T^{2}
73 1+(−36.5−63.2i)T2 1 + (-36.5 - 63.2i)T^{2}
79 1+(39.5−68.4i)T2 1 + (39.5 - 68.4i)T^{2}
83 1+9.48iT−83T2 1 + 9.48iT - 83T^{2}
89 1+(−15.4+8.94i)T+(44.5−77.0i)T2 1 + (-15.4 + 8.94i)T + (44.5 - 77.0i)T^{2}
97 1+97T2 1 + 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

−10.04257138277026615537867023048, −8.638293397901377234221662902362, −7.77745623132512787821592101051, −7.11627806544814219834165429313, −6.35771372127730143008455050305, −5.11709702014835413584820326101, −4.23967735770535923272289497703, −2.89753492467038051098543232863, −1.47974768009478107089592088843, −0.22621123577249492399583140723, 0.970759973161819766323144674124, 3.76904074401895361466268613771, 4.69719590798114491271384739260, 5.15399451658012768346845962023, 6.16794052114615529266807651925, 6.86503844653039864843620031909, 7.87961576661394101453849948722, 8.842778734784477504469237622865, 9.588496075028759604936999337936, 10.48232463752536450916394294136

Graph of the ZZ-function along the critical line