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Properties

Label 2-150-1.1-c7-0-5
Degree 22
Conductor 150150
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 27·3-s + 64·4-s − 216·6-s + 713·7-s − 512·8-s + 729·9-s + 3.81e3·11-s + 1.72e3·12-s − 391·13-s − 5.70e3·14-s + 4.09e3·16-s − 4.18e3·17-s − 5.83e3·18-s − 1.56e3·19-s + 1.92e4·21-s − 3.04e4·22-s + 1.14e5·23-s − 1.38e4·24-s + 3.12e3·26-s + 1.96e4·27-s + 4.56e4·28-s − 8.32e4·29-s − 8.31e4·31-s − 3.27e4·32-s + 1.02e5·33-s + 3.34e4·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.785·7-s − 0.353·8-s + 1/3·9-s + 0.863·11-s + 0.288·12-s − 0.0493·13-s − 0.555·14-s + 1/4·16-s − 0.206·17-s − 0.235·18-s − 0.0522·19-s + 0.453·21-s − 0.610·22-s + 1.95·23-s − 0.204·24-s + 0.0349·26-s + 0.192·27-s + 0.392·28-s − 0.633·29-s − 0.501·31-s − 0.176·32-s + 0.498·33-s + 0.145·34-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: 11
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: 00
Selberg data: (2, 150, ( :7/2), 1)(2,\ 150,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 2.2230888302.223088830
L(12)L(\frac12) ≈\approx 2.2230888302.223088830
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 1−713T+p7T2 1 - 713 T + p^{7} T^{2}
11 1−3810T+p7T2 1 - 3810 T + p^{7} T^{2}
13 1+391T+p7T2 1 + 391 T + p^{7} T^{2}
17 1+246pT+p7T2 1 + 246 p T + p^{7} T^{2}
19 1+1561T+p7T2 1 + 1561 T + p^{7} T^{2}
23 1−114150T+p7T2 1 - 114150 T + p^{7} T^{2}
29 1+83214T+p7T2 1 + 83214 T + p^{7} T^{2}
31 1+83167T+p7T2 1 + 83167 T + p^{7} T^{2}
37 1+231334T+p7T2 1 + 231334 T + p^{7} T^{2}
41 1+124656T+p7T2 1 + 124656 T + p^{7} T^{2}
43 1−193757T+p7T2 1 - 193757 T + p^{7} T^{2}
47 1−319290T+p7T2 1 - 319290 T + p^{7} T^{2}
53 1−1645428T+p7T2 1 - 1645428 T + p^{7} T^{2}
59 1+38610T+p7T2 1 + 38610 T + p^{7} T^{2}
61 1+1973905T+p7T2 1 + 1973905 T + p^{7} T^{2}
67 1−4409753T+p7T2 1 - 4409753 T + p^{7} T^{2}
71 1−124080T+p7T2 1 - 124080 T + p^{7} T^{2}
73 1−3967634T+p7T2 1 - 3967634 T + p^{7} T^{2}
79 1−7107992T+p7T2 1 - 7107992 T + p^{7} T^{2}
83 1−8117694T+p7T2 1 - 8117694 T + p^{7} T^{2}
89 1−6727872T+p7T2 1 - 6727872 T + p^{7} T^{2}
97 1+14268679T+p7T2 1 + 14268679 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

−11.45921194648592781667234084880, −10.66793854975150206855935899270, −9.339478086748993728805354471289, −8.740987090324215724318964116006, −7.62477996435617221779927777086, −6.68807695357198084146737698173, −5.06826054908479792590959942658, −3.59148035168922448000544851468, −2.11247490149974123584653563129, −0.970133117928985016681601604619, 0.970133117928985016681601604619, 2.11247490149974123584653563129, 3.59148035168922448000544851468, 5.06826054908479792590959942658, 6.68807695357198084146737698173, 7.62477996435617221779927777086, 8.740987090324215724318964116006, 9.339478086748993728805354471289, 10.66793854975150206855935899270, 11.45921194648592781667234084880

Graph of the ZZ-function along the critical line