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Properties

Label 2-150-1.1-c7-0-13
Degree 22
Conductor 150150
Sign −1-1
Analytic cond. 46.857746.8577
Root an. cond. 6.845276.84527
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 27·3-s + 64·4-s + 216·6-s + 1.12e3·7-s − 512·8-s + 729·9-s − 5.51e3·11-s − 1.72e3·12-s − 1.27e4·13-s − 9.00e3·14-s + 4.09e3·16-s + 3.22e4·17-s − 5.83e3·18-s − 4.44e3·19-s − 3.04e4·21-s + 4.41e4·22-s + 9.54e4·23-s + 1.38e4·24-s + 1.02e5·26-s − 1.96e4·27-s + 7.20e4·28-s + 1.94e4·29-s − 2.40e5·31-s − 3.27e4·32-s + 1.48e5·33-s − 2.57e5·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.24·7-s − 0.353·8-s + 1/3·9-s − 1.24·11-s − 0.288·12-s − 1.61·13-s − 0.877·14-s + 1/4·16-s + 1.58·17-s − 0.235·18-s − 0.148·19-s − 0.716·21-s + 0.883·22-s + 1.63·23-s + 0.204·24-s + 1.14·26-s − 0.192·27-s + 0.620·28-s + 0.148·29-s − 1.44·31-s − 0.176·32-s + 0.721·33-s − 1.12·34-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 46.857746.8577
Root analytic conductor: 6.845276.84527
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 150, ( :7/2), −1)(2,\ 150,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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+p3T 1 + p^{3} T
3 1+p3T 1 + p^{3} T
5 1 1
good7 1−1126T+p7T2 1 - 1126 T + p^{7} T^{2}
11 1+5518T+p7T2 1 + 5518 T + p^{7} T^{2}
13 1+12798T+p7T2 1 + 12798 T + p^{7} T^{2}
17 1−32206T+p7T2 1 - 32206 T + p^{7} T^{2}
19 1+4440T+p7T2 1 + 4440 T + p^{7} T^{2}
23 1−95452T+p7T2 1 - 95452 T + p^{7} T^{2}
29 1−19440T+p7T2 1 - 19440 T + p^{7} T^{2}
31 1+240248T+p7T2 1 + 240248 T + p^{7} T^{2}
37 1+77834T+p7T2 1 + 77834 T + p^{7} T^{2}
41 1−299522T+p7T2 1 - 299522 T + p^{7} T^{2}
43 1−416212T+p7T2 1 - 416212 T + p^{7} T^{2}
47 1−322976T+p7T2 1 - 322976 T + p^{7} T^{2}
53 1+880878T+p7T2 1 + 880878 T + p^{7} T^{2}
59 1+1845110T+p7T2 1 + 1845110 T + p^{7} T^{2}
61 1+861718T+p7T2 1 + 861718 T + p^{7} T^{2}
67 1+673864T+p7T2 1 + 673864 T + p^{7} T^{2}
71 1+3426948T+p7T2 1 + 3426948 T + p^{7} T^{2}
73 1+4678748T+p7T2 1 + 4678748 T + p^{7} T^{2}
79 1+3137760T+p7T2 1 + 3137760 T + p^{7} T^{2}
83 1−484132T+p7T2 1 - 484132 T + p^{7} T^{2}
89 1−6258710T+p7T2 1 - 6258710 T + p^{7} T^{2}
97 1−8657576T+p7T2 1 - 8657576 T + p^{7} T^{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

−10.98011576773272423000279665539, −10.32988064182692716188077073000, −9.186434659047989970153627051697, −7.74464548330772264926226270744, −7.38815030634840109665379961838, −5.54882663271926651205320758277, −4.84046004725109938798951389987, −2.74636154900655326346604413238, −1.36617533883969395014554057589, 0, 1.36617533883969395014554057589, 2.74636154900655326346604413238, 4.84046004725109938798951389987, 5.54882663271926651205320758277, 7.38815030634840109665379961838, 7.74464548330772264926226270744, 9.186434659047989970153627051697, 10.32988064182692716188077073000, 10.98011576773272423000279665539

Graph of the ZZ-function along the critical line