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Properties

Label 2-1260-105.53-c1-0-4
Degree 22
Conductor 12601260
Sign 0.416−0.909i0.416 - 0.909i
Analytic cond. 10.061110.0611
Root an. cond. 3.171933.17193
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  + (2.03 + 0.918i)5-s + (1.31 + 2.29i)7-s + (5.56 + 3.21i)11-s + (1.53 + 1.53i)13-s + (−3.67 + 0.985i)17-s + (−1.19 + 0.687i)19-s + (−8.15 − 2.18i)23-s + (3.31 + 3.74i)25-s − 5.95·29-s + (4.97 − 8.61i)31-s + (0.566 + 5.88i)35-s + (−2.39 − 0.642i)37-s − 1.18i·41-s + (3.28 + 3.28i)43-s + (0.655 − 2.44i)47-s + ⋯
L(s)  = 1  + (0.911 + 0.410i)5-s + (0.495 + 0.868i)7-s + (1.67 + 0.967i)11-s + (0.425 + 0.425i)13-s + (−0.892 + 0.239i)17-s + (−0.273 + 0.157i)19-s + (−1.70 − 0.455i)23-s + (0.662 + 0.748i)25-s − 1.10·29-s + (0.893 − 1.54i)31-s + (0.0957 + 0.995i)35-s + (−0.394 − 0.105i)37-s − 0.184i·41-s + (0.500 + 0.500i)43-s + (0.0955 − 0.356i)47-s + ⋯

Functional equation

Λ(s)=(1260s/2 ΓC(s) L(s)=((0.416−0.909i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1260s/2 ΓC(s+1/2) L(s)=((0.416−0.909i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12601260    =    22⋅32⋅5⋅72^{2} \cdot 3^{2} \cdot 5 \cdot 7
Sign: 0.416−0.909i0.416 - 0.909i
Analytic conductor: 10.061110.0611
Root analytic conductor: 3.171933.17193
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1260(53,⋅)\chi_{1260} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1260, ( :1/2), 0.416−0.909i)(2,\ 1260,\ (\ :1/2),\ 0.416 - 0.909i)

Particular Values

L(1)L(1) ≈\approx 2.1418663812.141866381
L(12)L(\frac12) ≈\approx 2.1418663812.141866381
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
3 1 1
5 1+(−2.03−0.918i)T 1 + (-2.03 - 0.918i)T
7 1+(−1.31−2.29i)T 1 + (-1.31 - 2.29i)T
good11 1+(−5.56−3.21i)T+(5.5+9.52i)T2 1 + (-5.56 - 3.21i)T + (5.5 + 9.52i)T^{2}
13 1+(−1.53−1.53i)T+13iT2 1 + (-1.53 - 1.53i)T + 13iT^{2}
17 1+(3.67−0.985i)T+(14.7−8.5i)T2 1 + (3.67 - 0.985i)T + (14.7 - 8.5i)T^{2}
19 1+(1.19−0.687i)T+(9.5−16.4i)T2 1 + (1.19 - 0.687i)T + (9.5 - 16.4i)T^{2}
23 1+(8.15+2.18i)T+(19.9+11.5i)T2 1 + (8.15 + 2.18i)T + (19.9 + 11.5i)T^{2}
29 1+5.95T+29T2 1 + 5.95T + 29T^{2}
31 1+(−4.97+8.61i)T+(−15.5−26.8i)T2 1 + (-4.97 + 8.61i)T + (-15.5 - 26.8i)T^{2}
37 1+(2.39+0.642i)T+(32.0+18.5i)T2 1 + (2.39 + 0.642i)T + (32.0 + 18.5i)T^{2}
41 1+1.18iT−41T2 1 + 1.18iT - 41T^{2}
43 1+(−3.28−3.28i)T+43iT2 1 + (-3.28 - 3.28i)T + 43iT^{2}
47 1+(−0.655+2.44i)T+(−40.7−23.5i)T2 1 + (-0.655 + 2.44i)T + (-40.7 - 23.5i)T^{2}
53 1+(−0.482−1.80i)T+(−45.8+26.5i)T2 1 + (-0.482 - 1.80i)T + (-45.8 + 26.5i)T^{2}
59 1+(3.27−5.67i)T+(−29.5−51.0i)T2 1 + (3.27 - 5.67i)T + (-29.5 - 51.0i)T^{2}
61 1+(−4.09−7.08i)T+(−30.5+52.8i)T2 1 + (-4.09 - 7.08i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.49+9.29i)T+(−58.0+33.5i)T2 1 + (2.49 + 9.29i)T + (-58.0 + 33.5i)T^{2}
71 1+8.06iT−71T2 1 + 8.06iT - 71T^{2}
73 1+(−12.9+3.47i)T+(63.2−36.5i)T2 1 + (-12.9 + 3.47i)T + (63.2 - 36.5i)T^{2}
79 1+(−8.00+4.62i)T+(39.5−68.4i)T2 1 + (-8.00 + 4.62i)T + (39.5 - 68.4i)T^{2}
83 1+(5.11−5.11i)T−83iT2 1 + (5.11 - 5.11i)T - 83iT^{2}
89 1+(3.97+6.88i)T+(−44.5+77.0i)T2 1 + (3.97 + 6.88i)T + (-44.5 + 77.0i)T^{2}
97 1+(0.537−0.537i)T−97iT2 1 + (0.537 - 0.537i)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

−9.596451752232788662192724785924, −9.220180414845295120566320617144, −8.376057602503914177684222852908, −7.26804963625991852504024090296, −6.20670438666246053804474802885, −6.05656609883075790730198934836, −4.62406763801639192182776396284, −3.88124461055396629407306025258, −2.22859278147549796168496882245, −1.77153501487396043615819465719, 0.963237080099873913766715364469, 1.93601786121330708314766830305, 3.53739981884527001171502870724, 4.29585710163474862662965233772, 5.38060776698330216658659287151, 6.28448149047572426527020864697, 6.85268728161917893595601662092, 8.123829455695954972989876567424, 8.745177154022593472555823342607, 9.491773569529085739838779941667

Graph of the ZZ-function along the critical line