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Properties

Label 2-260-20.3-c1-0-27
Degree 22
Conductor 260260
Sign 0.677+0.735i0.677 + 0.735i
Analytic cond. 2.076112.07611
Root an. cond. 1.440871.44087
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  + (−1.18 + 0.774i)2-s + (2.05 − 2.05i)3-s + (0.800 − 1.83i)4-s + (1.34 − 1.78i)5-s + (−0.841 + 4.03i)6-s + (0.845 + 0.845i)7-s + (0.472 + 2.78i)8-s − 5.48i·9-s + (−0.211 + 3.15i)10-s + 4.19i·11-s + (−2.12 − 5.42i)12-s + (−0.707 − 0.707i)13-s + (−1.65 − 0.345i)14-s + (−0.901 − 6.44i)15-s + (−2.71 − 2.93i)16-s + (−0.206 + 0.206i)17-s + ⋯
L(s)  = 1  + (−0.836 + 0.547i)2-s + (1.18 − 1.18i)3-s + (0.400 − 0.916i)4-s + (0.602 − 0.798i)5-s + (−0.343 + 1.64i)6-s + (0.319 + 0.319i)7-s + (0.167 + 0.985i)8-s − 1.82i·9-s + (−0.0668 + 0.997i)10-s + 1.26i·11-s + (−0.613 − 1.56i)12-s + (−0.196 − 0.196i)13-s + (−0.442 − 0.0923i)14-s + (−0.232 − 1.66i)15-s + (−0.679 − 0.733i)16-s + (−0.0499 + 0.0499i)17-s + ⋯

Functional equation

Λ(s)=(260s/2 ΓC(s) L(s)=((0.677+0.735i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(260s/2 ΓC(s+1/2) L(s)=((0.677+0.735i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 260260    =    22⋅5⋅132^{2} \cdot 5 \cdot 13
Sign: 0.677+0.735i0.677 + 0.735i
Analytic conductor: 2.076112.07611
Root analytic conductor: 1.440871.44087
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ260(183,⋅)\chi_{260} (183, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 260, ( :1/2), 0.677+0.735i)(2,\ 260,\ (\ :1/2),\ 0.677 + 0.735i)

Particular Values

L(1)L(1) ≈\approx 1.24186−0.544454i1.24186 - 0.544454i
L(12)L(\frac12) ≈\approx 1.24186−0.544454i1.24186 - 0.544454i
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+(1.18−0.774i)T 1 + (1.18 - 0.774i)T
5 1+(−1.34+1.78i)T 1 + (-1.34 + 1.78i)T
13 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good3 1+(−2.05+2.05i)T−3iT2 1 + (-2.05 + 2.05i)T - 3iT^{2}
7 1+(−0.845−0.845i)T+7iT2 1 + (-0.845 - 0.845i)T + 7iT^{2}
11 1−4.19iT−11T2 1 - 4.19iT - 11T^{2}
17 1+(0.206−0.206i)T−17iT2 1 + (0.206 - 0.206i)T - 17iT^{2}
19 1+4.84T+19T2 1 + 4.84T + 19T^{2}
23 1+(−0.389+0.389i)T−23iT2 1 + (-0.389 + 0.389i)T - 23iT^{2}
29 1−6.55iT−29T2 1 - 6.55iT - 29T^{2}
31 1−5.06iT−31T2 1 - 5.06iT - 31T^{2}
37 1+(−8.49+8.49i)T−37iT2 1 + (-8.49 + 8.49i)T - 37iT^{2}
41 1+5.61T+41T2 1 + 5.61T + 41T^{2}
43 1+(3.25−3.25i)T−43iT2 1 + (3.25 - 3.25i)T - 43iT^{2}
47 1+(−8.91−8.91i)T+47iT2 1 + (-8.91 - 8.91i)T + 47iT^{2}
53 1+(−0.0714−0.0714i)T+53iT2 1 + (-0.0714 - 0.0714i)T + 53iT^{2}
59 1+10.9T+59T2 1 + 10.9T + 59T^{2}
61 1+3.57T+61T2 1 + 3.57T + 61T^{2}
67 1+(6.95+6.95i)T+67iT2 1 + (6.95 + 6.95i)T + 67iT^{2}
71 1−4.61iT−71T2 1 - 4.61iT - 71T^{2}
73 1+(−2.57−2.57i)T+73iT2 1 + (-2.57 - 2.57i)T + 73iT^{2}
79 1−2.19T+79T2 1 - 2.19T + 79T^{2}
83 1+(−11.1+11.1i)T−83iT2 1 + (-11.1 + 11.1i)T - 83iT^{2}
89 1−15.0iT−89T2 1 - 15.0iT - 89T^{2}
97 1+(5.51−5.51i)T−97iT2 1 + (5.51 - 5.51i)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

−12.26526335771945314569916956975, −10.60792027796527485093444007985, −9.416072331899215513773520737807, −8.865220555353824039151006053897, −8.030341846394077544850694716319, −7.20393790862123756947945871050, −6.21690168681720357682758019704, −4.81789143000140728358159969750, −2.37950612741690324092583619719, −1.51448678024610146727083675230, 2.29401474144546655903742279797, 3.26173678437773459068906604756, 4.30977302605483677265833072611, 6.24018856946977272650101411055, 7.70251309358910595196092159449, 8.522245900310702297841898570733, 9.350067841253985573126931240136, 10.16981841571746133781286045287, 10.76597495198543245498758888440, 11.60261404042065375134296782694

Graph of the ZZ-function along the critical line