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Properties

Label 2-150-15.2-c7-0-27
Degree 22
Conductor 150150
Sign −0.134+0.990i-0.134 + 0.990i
Analytic cond. 46.857746.8577
Root an. cond. 6.845276.84527
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.65 − 5.65i)2-s + (−16.7 + 43.6i)3-s − 64.0i·4-s + (151. + 341. i)6-s + (469. + 469. i)7-s + (−362. − 362. i)8-s + (−1.62e3 − 1.46e3i)9-s − 2.97e3i·11-s + (2.79e3 + 1.07e3i)12-s + (−5.60e3 + 5.60e3i)13-s + 5.31e3·14-s − 4.09e3·16-s + (−7.32e3 + 7.32e3i)17-s + (−1.74e4 + 891. i)18-s − 4.78e4i·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.359 + 0.933i)3-s − 0.500i·4-s + (0.287 + 0.646i)6-s + (0.517 + 0.517i)7-s + (−0.250 − 0.250i)8-s + (−0.742 − 0.670i)9-s − 0.674i·11-s + (0.466 + 0.179i)12-s + (−0.708 + 0.708i)13-s + 0.517·14-s − 0.250·16-s + (−0.361 + 0.361i)17-s + (−0.706 + 0.0360i)18-s − 1.60i·19-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((−0.134+0.990i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+7/2) L(s)=((−0.134+0.990i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: −0.134+0.990i-0.134 + 0.990i
Analytic conductor: 46.857746.8577
Root analytic conductor: 6.845276.84527
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ150(107,⋅)\chi_{150} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :7/2), −0.134+0.990i)(2,\ 150,\ (\ :7/2),\ -0.134 + 0.990i)

Particular Values

L(4)L(4) ≈\approx 1.4592382481.459238248
L(12)L(\frac12) ≈\approx 1.4592382481.459238248
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+(−5.65+5.65i)T 1 + (-5.65 + 5.65i)T
3 1+(16.7−43.6i)T 1 + (16.7 - 43.6i)T
5 1 1
good7 1+(−469.−469.i)T+8.23e5iT2 1 + (-469. - 469. i)T + 8.23e5iT^{2}
11 1+2.97e3iT−1.94e7T2 1 + 2.97e3iT - 1.94e7T^{2}
13 1+(5.60e3−5.60e3i)T−6.27e7iT2 1 + (5.60e3 - 5.60e3i)T - 6.27e7iT^{2}
17 1+(7.32e3−7.32e3i)T−4.10e8iT2 1 + (7.32e3 - 7.32e3i)T - 4.10e8iT^{2}
19 1+4.78e4iT−8.93e8T2 1 + 4.78e4iT - 8.93e8T^{2}
23 1+(−4.77e4−4.77e4i)T+3.40e9iT2 1 + (-4.77e4 - 4.77e4i)T + 3.40e9iT^{2}
29 1+7.10e4T+1.72e10T2 1 + 7.10e4T + 1.72e10T^{2}
31 1−1.29e5T+2.75e10T2 1 - 1.29e5T + 2.75e10T^{2}
37 1+(4.20e5+4.20e5i)T+9.49e10iT2 1 + (4.20e5 + 4.20e5i)T + 9.49e10iT^{2}
41 1+1.82e5iT−1.94e11T2 1 + 1.82e5iT - 1.94e11T^{2}
43 1+(−2.70e5+2.70e5i)T−2.71e11iT2 1 + (-2.70e5 + 2.70e5i)T - 2.71e11iT^{2}
47 1+(−4.82e5+4.82e5i)T−5.06e11iT2 1 + (-4.82e5 + 4.82e5i)T - 5.06e11iT^{2}
53 1+(1.21e6+1.21e6i)T+1.17e12iT2 1 + (1.21e6 + 1.21e6i)T + 1.17e12iT^{2}
59 1−1.15e6T+2.48e12T2 1 - 1.15e6T + 2.48e12T^{2}
61 1−1.49e6T+3.14e12T2 1 - 1.49e6T + 3.14e12T^{2}
67 1+(6.72e5+6.72e5i)T+6.06e12iT2 1 + (6.72e5 + 6.72e5i)T + 6.06e12iT^{2}
71 1+4.36e6iT−9.09e12T2 1 + 4.36e6iT - 9.09e12T^{2}
73 1+(−3.38e6+3.38e6i)T−1.10e13iT2 1 + (-3.38e6 + 3.38e6i)T - 1.10e13iT^{2}
79 1−5.85e5iT−1.92e13T2 1 - 5.85e5iT - 1.92e13T^{2}
83 1+(5.25e6+5.25e6i)T+2.71e13iT2 1 + (5.25e6 + 5.25e6i)T + 2.71e13iT^{2}
89 1+9.78e6T+4.42e13T2 1 + 9.78e6T + 4.42e13T^{2}
97 1+(−9.89e6−9.89e6i)T+8.07e13iT2 1 + (-9.89e6 - 9.89e6i)T + 8.07e13iT^{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

−11.35606132048077852572123093019, −10.71731036774893848292256956853, −9.404673808119722748177660549541, −8.742288572984394324659434955839, −6.88445244927677627984762494731, −5.51218899405586522350326808377, −4.78034171666662849307681871397, −3.54969201763660988080819037405, −2.22089619110177217683046629324, −0.36006944503673612869477264624, 1.25403707936819888034976553888, 2.71607545236771331579096005254, 4.48006416713219160187973083848, 5.50053112024621358463789808786, 6.72046170921574969965721044008, 7.56268121465374817169039589900, 8.361047608911699085174989722880, 10.06501479679019981497971860606, 11.19860744686038034953103770701, 12.29922488709538304441136058323

Graph of the ZZ-function along the critical line