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Properties

Label 2-150-25.17-c2-0-0
Degree 22
Conductor 150150
Sign −0.527+0.849i-0.527 + 0.849i
Analytic cond. 4.087204.08720
Root an. cond. 2.021682.02168
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 + 1.26i)2-s + (−0.270 + 1.71i)3-s + (−1.17 − 1.61i)4-s + (−2.24 + 4.46i)5-s + (−1.98 − 1.43i)6-s + (−8.57 − 8.57i)7-s + (2.79 − 0.442i)8-s + (−2.85 − 0.927i)9-s + (−4.18 − 5.69i)10-s + (0.681 + 2.09i)11-s + (3.08 − 1.57i)12-s + (−0.642 − 1.26i)13-s + (16.3 − 5.30i)14-s + (−7.03 − 5.05i)15-s + (−1.23 + 3.80i)16-s + (−3.13 − 19.7i)17-s + ⋯
L(s)  = 1  + (−0.321 + 0.630i)2-s + (−0.0903 + 0.570i)3-s + (−0.293 − 0.404i)4-s + (−0.449 + 0.893i)5-s + (−0.330 − 0.239i)6-s + (−1.22 − 1.22i)7-s + (0.349 − 0.0553i)8-s + (−0.317 − 0.103i)9-s + (−0.418 − 0.569i)10-s + (0.0619 + 0.190i)11-s + (0.257 − 0.131i)12-s + (−0.0494 − 0.0969i)13-s + (1.16 − 0.378i)14-s + (−0.468 − 0.336i)15-s + (−0.0772 + 0.237i)16-s + (−0.184 − 1.16i)17-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((−0.527+0.849i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+1) L(s)=((−0.527+0.849i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: −0.527+0.849i-0.527 + 0.849i
Analytic conductor: 4.087204.08720
Root analytic conductor: 2.021682.02168
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ150(67,⋅)\chi_{150} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :1), −0.527+0.849i)(2,\ 150,\ (\ :1),\ -0.527 + 0.849i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0391952−0.0704440i0.0391952 - 0.0704440i
L(12)L(\frac12) ≈\approx 0.0391952−0.0704440i0.0391952 - 0.0704440i
L(2)L(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+(0.642−1.26i)T 1 + (0.642 - 1.26i)T
3 1+(0.270−1.71i)T 1 + (0.270 - 1.71i)T
5 1+(2.24−4.46i)T 1 + (2.24 - 4.46i)T
good7 1+(8.57+8.57i)T+49iT2 1 + (8.57 + 8.57i)T + 49iT^{2}
11 1+(−0.681−2.09i)T+(−97.8+71.1i)T2 1 + (-0.681 - 2.09i)T + (-97.8 + 71.1i)T^{2}
13 1+(0.642+1.26i)T+(−99.3+136.i)T2 1 + (0.642 + 1.26i)T + (-99.3 + 136. i)T^{2}
17 1+(3.13+19.7i)T+(−274.+89.3i)T2 1 + (3.13 + 19.7i)T + (-274. + 89.3i)T^{2}
19 1+(14.3−19.7i)T+(−111.−343.i)T2 1 + (14.3 - 19.7i)T + (-111. - 343. i)T^{2}
23 1+(21.3+10.8i)T+(310.+427.i)T2 1 + (21.3 + 10.8i)T + (310. + 427. i)T^{2}
29 1+(−22.2−30.5i)T+(−259.+799.i)T2 1 + (-22.2 - 30.5i)T + (-259. + 799. i)T^{2}
31 1+(23.8+17.3i)T+(296.+913.i)T2 1 + (23.8 + 17.3i)T + (296. + 913. i)T^{2}
37 1+(7.25−3.69i)T+(804.−1.10e3i)T2 1 + (7.25 - 3.69i)T + (804. - 1.10e3i)T^{2}
41 1+(21.4−66.0i)T+(−1.35e3−988.i)T2 1 + (21.4 - 66.0i)T + (-1.35e3 - 988. i)T^{2}
43 1+(26.5−26.5i)T−1.84e3iT2 1 + (26.5 - 26.5i)T - 1.84e3iT^{2}
47 1+(16.5+2.62i)T+(2.10e3+682.i)T2 1 + (16.5 + 2.62i)T + (2.10e3 + 682. i)T^{2}
53 1+(12.8−81.3i)T+(−2.67e3−868.i)T2 1 + (12.8 - 81.3i)T + (-2.67e3 - 868. i)T^{2}
59 1+(−27.3−8.87i)T+(2.81e3+2.04e3i)T2 1 + (-27.3 - 8.87i)T + (2.81e3 + 2.04e3i)T^{2}
61 1+(20.9+64.5i)T+(−3.01e3+2.18e3i)T2 1 + (20.9 + 64.5i)T + (-3.01e3 + 2.18e3i)T^{2}
67 1+(1.94+12.2i)T+(−4.26e3+1.38e3i)T2 1 + (1.94 + 12.2i)T + (-4.26e3 + 1.38e3i)T^{2}
71 1+(−97.9+71.1i)T+(1.55e3−4.79e3i)T2 1 + (-97.9 + 71.1i)T + (1.55e3 - 4.79e3i)T^{2}
73 1+(−91.3−46.5i)T+(3.13e3+4.31e3i)T2 1 + (-91.3 - 46.5i)T + (3.13e3 + 4.31e3i)T^{2}
79 1+(66.9+92.1i)T+(−1.92e3+5.93e3i)T2 1 + (66.9 + 92.1i)T + (-1.92e3 + 5.93e3i)T^{2}
83 1+(78.3−12.4i)T+(6.55e3−2.12e3i)T2 1 + (78.3 - 12.4i)T + (6.55e3 - 2.12e3i)T^{2}
89 1+(0.353−0.114i)T+(6.40e3−4.65e3i)T2 1 + (0.353 - 0.114i)T + (6.40e3 - 4.65e3i)T^{2}
97 1+(86.8+13.7i)T+(8.94e3+2.90e3i)T2 1 + (86.8 + 13.7i)T + (8.94e3 + 2.90e3i)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

−13.77203028634661946513955813371, −12.48853394644022143591528803484, −11.07161855915661609055296321528, −10.20686344897159226493447781080, −9.625348808805223683913142366250, −8.062669170527813982235619622270, −6.98578225342290777074209245167, −6.25262695480759975650578689633, −4.42264958819874127346180811318, −3.28351207556898226771520406338, 0.05511945933139242973575086440, 2.15572870001148366073556301974, 3.76130156729666560227485937279, 5.48757686007596681307284545907, 6.69735553297371403286526489470, 8.331304357952991955056482463783, 8.871629545752914034058662426982, 9.961259248935907959272792243075, 11.38026015326546110254108190575, 12.29109845773352126167756416491

Graph of the ZZ-function along the critical line