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Properties

Label 2-980-140.107-c1-0-111
Degree 22
Conductor 980980
Sign 0.0134−0.999i0.0134 - 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  + (0.604 − 1.27i)2-s + (0.844 − 3.15i)3-s + (−1.26 − 1.54i)4-s + (−1.79 + 1.33i)5-s + (−3.52 − 2.98i)6-s + (−2.74 + 0.689i)8-s + (−6.62 − 3.82i)9-s + (0.624 + 3.10i)10-s + (−1.96 + 1.13i)11-s + (−5.94 + 2.69i)12-s + (1.38 − 1.38i)13-s + (2.69 + 6.78i)15-s + (−0.775 + 3.92i)16-s + (0.0499 − 0.186i)17-s + (−8.89 + 6.15i)18-s + (3.45 − 5.98i)19-s + ⋯
L(s)  = 1  + (0.427 − 0.904i)2-s + (0.487 − 1.81i)3-s + (−0.634 − 0.772i)4-s + (−0.801 + 0.597i)5-s + (−1.43 − 1.21i)6-s + (−0.969 + 0.243i)8-s + (−2.20 − 1.27i)9-s + (0.197 + 0.980i)10-s + (−0.593 + 0.342i)11-s + (−1.71 + 0.778i)12-s + (0.383 − 0.383i)13-s + (0.696 + 1.75i)15-s + (−0.193 + 0.981i)16-s + (0.0121 − 0.0451i)17-s + (−2.09 + 1.45i)18-s + (0.792 − 1.37i)19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.0134−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0134 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.0134−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0134 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.0134−0.999i0.0134 - 0.999i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(667,⋅)\chi_{980} (667, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.0134−0.999i)(2,\ 980,\ (\ :1/2),\ 0.0134 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.679484+0.670403i0.679484 + 0.670403i
L(12)L(\frac12) ≈\approx 0.679484+0.670403i0.679484 + 0.670403i
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+(−0.604+1.27i)T 1 + (-0.604 + 1.27i)T
5 1+(1.79−1.33i)T 1 + (1.79 - 1.33i)T
7 1 1
good3 1+(−0.844+3.15i)T+(−2.59−1.5i)T2 1 + (-0.844 + 3.15i)T + (-2.59 - 1.5i)T^{2}
11 1+(1.96−1.13i)T+(5.5−9.52i)T2 1 + (1.96 - 1.13i)T + (5.5 - 9.52i)T^{2}
13 1+(−1.38+1.38i)T−13iT2 1 + (-1.38 + 1.38i)T - 13iT^{2}
17 1+(−0.0499+0.186i)T+(−14.7−8.5i)T2 1 + (-0.0499 + 0.186i)T + (-14.7 - 8.5i)T^{2}
19 1+(−3.45+5.98i)T+(−9.5−16.4i)T2 1 + (-3.45 + 5.98i)T + (-9.5 - 16.4i)T^{2}
23 1+(3.23−0.866i)T+(19.9−11.5i)T2 1 + (3.23 - 0.866i)T + (19.9 - 11.5i)T^{2}
29 1−7.33iT−29T2 1 - 7.33iT - 29T^{2}
31 1+(−0.430+0.248i)T+(15.5−26.8i)T2 1 + (-0.430 + 0.248i)T + (15.5 - 26.8i)T^{2}
37 1+(3.37−0.904i)T+(32.0−18.5i)T2 1 + (3.37 - 0.904i)T + (32.0 - 18.5i)T^{2}
41 1−3.22T+41T2 1 - 3.22T + 41T^{2}
43 1+(2.91+2.91i)T+43iT2 1 + (2.91 + 2.91i)T + 43iT^{2}
47 1+(−0.645−2.40i)T+(−40.7+23.5i)T2 1 + (-0.645 - 2.40i)T + (-40.7 + 23.5i)T^{2}
53 1+(6.98+1.87i)T+(45.8+26.5i)T2 1 + (6.98 + 1.87i)T + (45.8 + 26.5i)T^{2}
59 1+(2.61+4.52i)T+(−29.5+51.0i)T2 1 + (2.61 + 4.52i)T + (-29.5 + 51.0i)T^{2}
61 1+(−5.00+8.67i)T+(−30.5−52.8i)T2 1 + (-5.00 + 8.67i)T + (-30.5 - 52.8i)T^{2}
67 1+(12.0+3.21i)T+(58.0+33.5i)T2 1 + (12.0 + 3.21i)T + (58.0 + 33.5i)T^{2}
71 1+3.60iT−71T2 1 + 3.60iT - 71T^{2}
73 1+(12.6+3.38i)T+(63.2+36.5i)T2 1 + (12.6 + 3.38i)T + (63.2 + 36.5i)T^{2}
79 1+(−5.66+9.81i)T+(−39.5−68.4i)T2 1 + (-5.66 + 9.81i)T + (-39.5 - 68.4i)T^{2}
83 1+(0.591+0.591i)T+83iT2 1 + (0.591 + 0.591i)T + 83iT^{2}
89 1+(3.45+1.99i)T+(44.5+77.0i)T2 1 + (3.45 + 1.99i)T + (44.5 + 77.0i)T^{2}
97 1+(1.09+1.09i)T+97iT2 1 + (1.09 + 1.09i)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

−9.264550238492010029655755207966, −8.440778969431994352452953698159, −7.63631636554058222270904499904, −6.94402107117828579395664128910, −6.05601524718866711485939918563, −4.91215224170800187082074617035, −3.36249491526999453495217016559, −2.84628671728495739761212721107, −1.71390309052939050977138259148, −0.35358893876480299876672390923, 2.98062853792938768110717225823, 3.89603894392084962993998478234, 4.33403183763944902983486059404, 5.32144328079970953224649648028, 5.95640608059892687475044447794, 7.62825658349397665832690122735, 8.206643222819658670993523655631, 8.812564864307479162600329930521, 9.634940186825094571767712559371, 10.36083258518905835173100188248

Graph of the ZZ-function along the critical line