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Properties

Label 2-150-5.4-c7-0-11
Degree 22
Conductor 150150
Sign 0.894+0.447i0.894 + 0.447i
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  − 8i·2-s + 27i·3-s − 64·4-s + 216·6-s + 391i·7-s + 512i·8-s − 729·9-s − 4.39e3·11-s − 1.72e3i·12-s − 1.34e4i·13-s + 3.12e3·14-s + 4.09e3·16-s + 7.68e3i·17-s + 5.83e3i·18-s + 1.37e4·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.430i·7-s + 0.353i·8-s − 0.333·9-s − 0.996·11-s − 0.288i·12-s − 1.69i·13-s + 0.304·14-s + 0.250·16-s + 0.379i·17-s + 0.235i·18-s + 0.458·19-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+7/2) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: 0.894+0.447i0.894 + 0.447i
Analytic conductor: 46.857746.8577
Root analytic conductor: 6.845276.84527
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ150(49,⋅)\chi_{150} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :7/2), 0.894+0.447i)(2,\ 150,\ (\ :7/2),\ 0.894 + 0.447i)

Particular Values

L(4)L(4) ≈\approx 1.6188347361.618834736
L(12)L(\frac12) ≈\approx 1.6188347361.618834736
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+8iT 1 + 8iT
3 1−27iT 1 - 27iT
5 1 1
good7 1−391iT−8.23e5T2 1 - 391iT - 8.23e5T^{2}
11 1+4.39e3T+1.94e7T2 1 + 4.39e3T + 1.94e7T^{2}
13 1+1.34e4iT−6.27e7T2 1 + 1.34e4iT - 6.27e7T^{2}
17 1−7.68e3iT−4.10e8T2 1 - 7.68e3iT - 4.10e8T^{2}
19 1−1.37e4T+8.93e8T2 1 - 1.37e4T + 8.93e8T^{2}
23 1−3.54e4iT−3.40e9T2 1 - 3.54e4iT - 3.40e9T^{2}
29 1−1.57e5T+1.72e10T2 1 - 1.57e5T + 1.72e10T^{2}
31 1+9.93e4T+2.75e10T2 1 + 9.93e4T + 2.75e10T^{2}
37 1−1.61e5iT−9.49e10T2 1 - 1.61e5iT - 9.49e10T^{2}
41 1−5.21e5T+1.94e11T2 1 - 5.21e5T + 1.94e11T^{2}
43 1−3.40e5iT−2.71e11T2 1 - 3.40e5iT - 2.71e11T^{2}
47 1−5.08e4iT−5.06e11T2 1 - 5.08e4iT - 5.06e11T^{2}
53 1+8.91e5iT−1.17e12T2 1 + 8.91e5iT - 1.17e12T^{2}
59 1−1.34e6T+2.48e12T2 1 - 1.34e6T + 2.48e12T^{2}
61 1−3.39e6T+3.14e12T2 1 - 3.39e6T + 3.14e12T^{2}
67 1−2.24e6iT−6.06e12T2 1 - 2.24e6iT - 6.06e12T^{2}
71 1−2.73e6T+9.09e12T2 1 - 2.73e6T + 9.09e12T^{2}
73 1+5.02e6iT−1.10e13T2 1 + 5.02e6iT - 1.10e13T^{2}
79 1+1.57e6T+1.92e13T2 1 + 1.57e6T + 1.92e13T^{2}
83 1+7.79e6iT−2.71e13T2 1 + 7.79e6iT - 2.71e13T^{2}
89 1−5.80e6T+4.42e13T2 1 - 5.80e6T + 4.42e13T^{2}
97 1−2.49e6iT−8.07e13T2 1 - 2.49e6iT - 8.07e13T^{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

−11.48519671570809561493621341679, −10.49337007390083128974833537127, −9.896978957823366808642658464928, −8.624979881973633056590674474981, −7.73475764715182246712930746004, −5.77619748568146310290896091864, −4.97291923013368513672036060904, −3.45130182480188653390703064844, −2.50810715866138551598039632187, −0.69533343277918214669614747826, 0.73982100135571957176577749057, 2.37418218062608443963522803983, 4.12644510167032123690826295023, 5.34760769385700164135427999730, 6.65218933767494223036784120148, 7.36339014793102259309513929543, 8.453387167078567504802141829034, 9.504950163020247138574651077647, 10.75026838569408361694920302226, 11.90677760786218491724298961118

Graph of the ZZ-function along the critical line