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Properties

Label 2-1470-35.13-c1-0-26
Degree 22
Conductor 14701470
Sign −0.333+0.942i-0.333 + 0.942i
Analytic cond. 11.738011.7380
Root an. cond. 3.426073.42607
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 + (−0.707 + 0.707i)3-s − 1.00i·4-s + (0.461 − 2.18i)5-s − 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (1.22 + 1.87i)10-s − 2.98·11-s + (0.707 + 0.707i)12-s + (−0.960 + 0.960i)13-s + (1.22 + 1.87i)15-s − 1.00·16-s + (−1.62 − 1.62i)17-s + (0.707 + 0.707i)18-s + 8.67·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.206 − 0.978i)5-s − 0.408i·6-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.385 + 0.592i)10-s − 0.899·11-s + (0.204 + 0.204i)12-s + (−0.266 + 0.266i)13-s + (0.315 + 0.483i)15-s − 0.250·16-s + (−0.394 − 0.394i)17-s + (0.166 + 0.166i)18-s + 1.98·19-s + ⋯

Functional equation

Λ(s)=(1470s/2 ΓC(s) L(s)=((−0.333+0.942i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1470s/2 ΓC(s+1/2) L(s)=((−0.333+0.942i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14701470    =    2⋅3⋅5⋅722 \cdot 3 \cdot 5 \cdot 7^{2}
Sign: −0.333+0.942i-0.333 + 0.942i
Analytic conductor: 11.738011.7380
Root analytic conductor: 3.426073.42607
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1470(1273,⋅)\chi_{1470} (1273, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1470, ( :1/2), −0.333+0.942i)(2,\ 1470,\ (\ :1/2),\ -0.333 + 0.942i)

Particular Values

L(1)L(1) ≈\approx 0.47498933960.4749893396
L(12)L(\frac12) ≈\approx 0.47498933960.4749893396
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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
5 1+(−0.461+2.18i)T 1 + (-0.461 + 2.18i)T
7 1 1
good11 1+2.98T+11T2 1 + 2.98T + 11T^{2}
13 1+(0.960−0.960i)T−13iT2 1 + (0.960 - 0.960i)T - 13iT^{2}
17 1+(1.62+1.62i)T+17iT2 1 + (1.62 + 1.62i)T + 17iT^{2}
19 1−8.67T+19T2 1 - 8.67T + 19T^{2}
23 1+(−1.36−1.36i)T+23iT2 1 + (-1.36 - 1.36i)T + 23iT^{2}
29 1+2.00iT−29T2 1 + 2.00iT - 29T^{2}
31 1+0.179iT−31T2 1 + 0.179iT - 31T^{2}
37 1+(4.86−4.86i)T−37iT2 1 + (4.86 - 4.86i)T - 37iT^{2}
41 1+5.14iT−41T2 1 + 5.14iT - 41T^{2}
43 1+(7.01+7.01i)T+43iT2 1 + (7.01 + 7.01i)T + 43iT^{2}
47 1+(−0.202−0.202i)T+47iT2 1 + (-0.202 - 0.202i)T + 47iT^{2}
53 1+(7.01+7.01i)T+53iT2 1 + (7.01 + 7.01i)T + 53iT^{2}
59 1+7.09T+59T2 1 + 7.09T + 59T^{2}
61 1+2.41iT−61T2 1 + 2.41iT - 61T^{2}
67 1+(6.29−6.29i)T−67iT2 1 + (6.29 - 6.29i)T - 67iT^{2}
71 1+9.08T+71T2 1 + 9.08T + 71T^{2}
73 1+(8.78−8.78i)T−73iT2 1 + (8.78 - 8.78i)T - 73iT^{2}
79 1+16.2iT−79T2 1 + 16.2iT - 79T^{2}
83 1+(−8.11+8.11i)T−83iT2 1 + (-8.11 + 8.11i)T - 83iT^{2}
89 1+12.3T+89T2 1 + 12.3T + 89T^{2}
97 1+(−7.44−7.44i)T+97iT2 1 + (-7.44 - 7.44i)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

−9.290043716010047046205633037558, −8.560063910631368598060637591167, −7.67246945486024083120526500143, −6.95703160331051175072458271695, −5.75812053830462583663203862397, −5.21066512370421166720321395060, −4.57291448363719071077750085292, −3.15782824628179088702868965640, −1.62334109794843661474721453982, −0.24092409628550973267663530484, 1.44891190248953267679651567613, 2.68370379698863319864645552367, 3.31960584419453372403274072222, 4.81014766791964966200606753208, 5.71400716031413787959689353109, 6.65409428101839260309547922167, 7.49242974771367992072408173405, 7.917641887110297045947388214100, 9.123515916078323181395069198016, 9.916205598794640671942418053740

Graph of the ZZ-function along the critical line