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Properties

Label 2-980-20.3-c1-0-21
Degree 22
Conductor 980980
Sign −0.996+0.0781i-0.996 + 0.0781i
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  + (0.566 + 1.29i)2-s + (0.792 − 0.792i)3-s + (−1.35 + 1.46i)4-s + (−0.427 + 2.19i)5-s + (1.47 + 0.578i)6-s + (−2.67 − 0.928i)8-s + 1.74i·9-s + (−3.08 + 0.689i)10-s + 2.98i·11-s + (0.0868 + 2.24i)12-s + (−4.05 − 4.05i)13-s + (1.40 + 2.07i)15-s + (−0.309 − 3.98i)16-s + (−1.68 + 1.68i)17-s + (−2.25 + 0.987i)18-s + 2.51·19-s + ⋯
L(s)  = 1  + (0.400 + 0.916i)2-s + (0.457 − 0.457i)3-s + (−0.679 + 0.733i)4-s + (−0.191 + 0.981i)5-s + (0.602 + 0.236i)6-s + (−0.944 − 0.328i)8-s + 0.581i·9-s + (−0.975 + 0.218i)10-s + 0.899i·11-s + (0.0250 + 0.646i)12-s + (−1.12 − 1.12i)13-s + (0.361 + 0.536i)15-s + (−0.0774 − 0.996i)16-s + (−0.409 + 0.409i)17-s + (−0.532 + 0.232i)18-s + 0.576·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.996+0.0781i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0781i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.996+0.0781i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0781i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.996+0.0781i-0.996 + 0.0781i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(883,⋅)\chi_{980} (883, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.996+0.0781i)(2,\ 980,\ (\ :1/2),\ -0.996 + 0.0781i)

Particular Values

L(1)L(1) ≈\approx 0.0507461−1.29593i0.0507461 - 1.29593i
L(12)L(\frac12) ≈\approx 0.0507461−1.29593i0.0507461 - 1.29593i
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+(−0.566−1.29i)T 1 + (-0.566 - 1.29i)T
5 1+(0.427−2.19i)T 1 + (0.427 - 2.19i)T
7 1 1
good3 1+(−0.792+0.792i)T−3iT2 1 + (-0.792 + 0.792i)T - 3iT^{2}
11 1−2.98iT−11T2 1 - 2.98iT - 11T^{2}
13 1+(4.05+4.05i)T+13iT2 1 + (4.05 + 4.05i)T + 13iT^{2}
17 1+(1.68−1.68i)T−17iT2 1 + (1.68 - 1.68i)T - 17iT^{2}
19 1−2.51T+19T2 1 - 2.51T + 19T^{2}
23 1+(4.24−4.24i)T−23iT2 1 + (4.24 - 4.24i)T - 23iT^{2}
29 1+2.55iT−29T2 1 + 2.55iT - 29T^{2}
31 1+3.60iT−31T2 1 + 3.60iT - 31T^{2}
37 1+(3.37−3.37i)T−37iT2 1 + (3.37 - 3.37i)T - 37iT^{2}
41 1−7.93T+41T2 1 - 7.93T + 41T^{2}
43 1+(7.62−7.62i)T−43iT2 1 + (7.62 - 7.62i)T - 43iT^{2}
47 1+(−2.09−2.09i)T+47iT2 1 + (-2.09 - 2.09i)T + 47iT^{2}
53 1+(−1.80−1.80i)T+53iT2 1 + (-1.80 - 1.80i)T + 53iT^{2}
59 1+2.09T+59T2 1 + 2.09T + 59T^{2}
61 1−1.90T+61T2 1 - 1.90T + 61T^{2}
67 1+(−2.19−2.19i)T+67iT2 1 + (-2.19 - 2.19i)T + 67iT^{2}
71 1−8.09iT−71T2 1 - 8.09iT - 71T^{2}
73 1+(6.89+6.89i)T+73iT2 1 + (6.89 + 6.89i)T + 73iT^{2}
79 1−8.06T+79T2 1 - 8.06T + 79T^{2}
83 1+(5.99−5.99i)T−83iT2 1 + (5.99 - 5.99i)T - 83iT^{2}
89 1−2.05iT−89T2 1 - 2.05iT - 89T^{2}
97 1+(6.63−6.63i)T−97iT2 1 + (6.63 - 6.63i)T - 97iT^{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

−10.14882507451474171544963660576, −9.645798249517228397041450142339, −8.302960617383561221460695186816, −7.57311416914847572379063103689, −7.37375935464770803031489583445, −6.29105404216472065627567736853, −5.33606274647357604086243932040, −4.34674760103215815846867199700, −3.15893025118583444692328531810, −2.27425719278006456878514297540, 0.47345582125362518125314857265, 2.02150358921523590130496215833, 3.24614625655133590079894831973, 4.15626929912051175038157092204, 4.82955919511337001483418454943, 5.80190045526627979957936189303, 6.97447759668994268224265150574, 8.407334515206537201719180557018, 8.965500630050285345290126184829, 9.529151177475322206247600675704

Graph of the ZZ-function along the critical line