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Properties

Label 2-1350-15.2-c1-0-15
Degree 22
Conductor 13501350
Sign 0.608+0.793i0.608 + 0.793i
Analytic cond. 10.779810.7798
Root an. cond. 3.283263.28326
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 + 0.707i)2-s − 1.00i·4-s + (0.896 + 0.896i)7-s + (0.707 + 0.707i)8-s − 3i·11-s + (2.12 − 2.12i)13-s − 1.26·14-s − 1.00·16-s + (−1.55 + 1.55i)17-s − 6.19i·19-s + (2.12 + 2.12i)22-s + (−2.12 − 2.12i)23-s + 3i·26-s + (0.896 − 0.896i)28-s − 8.19·29-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.338 + 0.338i)7-s + (0.250 + 0.250i)8-s − 0.904i·11-s + (0.588 − 0.588i)13-s − 0.338·14-s − 0.250·16-s + (−0.376 + 0.376i)17-s − 1.42i·19-s + (0.452 + 0.452i)22-s + (−0.442 − 0.442i)23-s + 0.588i·26-s + (0.169 − 0.169i)28-s − 1.52·29-s + ⋯

Functional equation

Λ(s)=(1350s/2 ΓC(s) L(s)=((0.608+0.793i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1350s/2 ΓC(s+1/2) L(s)=((0.608+0.793i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13501350    =    2⋅33⋅522 \cdot 3^{3} \cdot 5^{2}
Sign: 0.608+0.793i0.608 + 0.793i
Analytic conductor: 10.779810.7798
Root analytic conductor: 3.283263.28326
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1350(107,⋅)\chi_{1350} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1350, ( :1/2), 0.608+0.793i)(2,\ 1350,\ (\ :1/2),\ 0.608 + 0.793i)

Particular Values

L(1)L(1) ≈\approx 1.0630911141.063091114
L(12)L(\frac12) ≈\approx 1.0630911141.063091114
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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
3 1 1
5 1 1
good7 1+(−0.896−0.896i)T+7iT2 1 + (-0.896 - 0.896i)T + 7iT^{2}
11 1+3iT−11T2 1 + 3iT - 11T^{2}
13 1+(−2.12+2.12i)T−13iT2 1 + (-2.12 + 2.12i)T - 13iT^{2}
17 1+(1.55−1.55i)T−17iT2 1 + (1.55 - 1.55i)T - 17iT^{2}
19 1+6.19iT−19T2 1 + 6.19iT - 19T^{2}
23 1+(2.12+2.12i)T+23iT2 1 + (2.12 + 2.12i)T + 23iT^{2}
29 1+8.19T+29T2 1 + 8.19T + 29T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1+(−7.02−7.02i)T+37iT2 1 + (-7.02 - 7.02i)T + 37iT^{2}
41 1+6iT−41T2 1 + 6iT - 41T^{2}
43 1+(−7.58+7.58i)T−43iT2 1 + (-7.58 + 7.58i)T - 43iT^{2}
47 1+(6.36−6.36i)T−47iT2 1 + (6.36 - 6.36i)T - 47iT^{2}
53 1+(1.55+1.55i)T+53iT2 1 + (1.55 + 1.55i)T + 53iT^{2}
59 1−13.3T+59T2 1 - 13.3T + 59T^{2}
61 1+9.19T+61T2 1 + 9.19T + 61T^{2}
67 1+(1.55+1.55i)T+67iT2 1 + (1.55 + 1.55i)T + 67iT^{2}
71 1+0.803iT−71T2 1 + 0.803iT - 71T^{2}
73 1+(−6.03+6.03i)T−73iT2 1 + (-6.03 + 6.03i)T - 73iT^{2}
79 1+10.1iT−79T2 1 + 10.1iT - 79T^{2}
83 1+(3.10+3.10i)T+83iT2 1 + (3.10 + 3.10i)T + 83iT^{2}
89 1−8.19T+89T2 1 - 8.19T + 89T^{2}
97 1+(1.88+1.88i)T+97iT2 1 + (1.88 + 1.88i)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

−9.209840672968511034700763846601, −8.681977600822778502233185994896, −7.993476278119332052353142210911, −7.11640058943353434235265913545, −6.14339681641352935181771131464, −5.56106862724367152994134598283, −4.51833731289144488279841802425, −3.30882528195806494555472071147, −2.05410301838966525042489425367, −0.54242050434655473379279638548, 1.37629595288804470575697488086, 2.28088967688965539582230508217, 3.73642977488491913422420120324, 4.32616727665330000030499225190, 5.56327617868676416663734056207, 6.57387243147352117944930640091, 7.58969153828471736454559839989, 7.991754322337710405177645027959, 9.211167318467304992941910061115, 9.594088760668680366669842124382

Graph of the ZZ-function along the critical line