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Properties

Label 2-152-152.67-c3-0-11
Degree 22
Conductor 152152
Sign −0.657−0.753i-0.657 - 0.753i
Analytic cond. 8.968298.96829
Root an. cond. 2.994712.99471
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.78 + 0.491i)2-s + (6.01 + 7.16i)3-s + (7.51 − 2.73i)4-s + (−20.2 − 17.0i)6-s + (−19.5 + 11.3i)8-s + (−10.5 + 59.6i)9-s + (15.8 + 27.4i)11-s + (64.8 + 37.4i)12-s + (49.0 − 41.1i)16-s + (8.34 + 47.3i)17-s − 171. i·18-s + (−71.5 − 41.6i)19-s + (−57.5 − 68.6i)22-s + (−198. − 72.4i)24-s + (95.7 + 80.3i)25-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (1.15 + 1.37i)3-s + (0.939 − 0.342i)4-s + (−1.37 − 1.15i)6-s + (−0.866 + 0.500i)8-s + (−0.389 + 2.20i)9-s + (0.433 + 0.751i)11-s + (1.55 + 0.900i)12-s + (0.766 − 0.642i)16-s + (0.119 + 0.675i)17-s − 2.24i·18-s + (−0.864 − 0.503i)19-s + (−0.557 − 0.664i)22-s + (−1.69 − 0.615i)24-s + (0.766 + 0.642i)25-s + ⋯

Functional equation

Λ(s)=(152s/2 ΓC(s) L(s)=((−0.657−0.753i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(152s/2 ΓC(s+3/2) L(s)=((−0.657−0.753i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 152152    =    23⋅192^{3} \cdot 19
Sign: −0.657−0.753i-0.657 - 0.753i
Analytic conductor: 8.968298.96829
Root analytic conductor: 2.994712.99471
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ152(67,⋅)\chi_{152} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 152, ( :3/2), −0.657−0.753i)(2,\ 152,\ (\ :3/2),\ -0.657 - 0.753i)

Particular Values

L(2)L(2) ≈\approx 0.608313+1.33839i0.608313 + 1.33839i
L(12)L(\frac12) ≈\approx 0.608313+1.33839i0.608313 + 1.33839i
L(52)L(\frac{5}{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+(2.78−0.491i)T 1 + (2.78 - 0.491i)T
19 1+(71.5+41.6i)T 1 + (71.5 + 41.6i)T
good3 1+(−6.01−7.16i)T+(−4.68+26.5i)T2 1 + (-6.01 - 7.16i)T + (-4.68 + 26.5i)T^{2}
5 1+(−95.7−80.3i)T2 1 + (-95.7 - 80.3i)T^{2}
7 1+(171.5+297.i)T2 1 + (171.5 + 297. i)T^{2}
11 1+(−15.8−27.4i)T+(−665.5+1.15e3i)T2 1 + (-15.8 - 27.4i)T + (-665.5 + 1.15e3i)T^{2}
13 1+(381.+2.16e3i)T2 1 + (381. + 2.16e3i)T^{2}
17 1+(−8.34−47.3i)T+(−4.61e3+1.68e3i)T2 1 + (-8.34 - 47.3i)T + (-4.61e3 + 1.68e3i)T^{2}
23 1+(−9.32e3+7.82e3i)T2 1 + (-9.32e3 + 7.82e3i)T^{2}
29 1+(−2.29e4−8.34e3i)T2 1 + (-2.29e4 - 8.34e3i)T^{2}
31 1+(−1.48e4−2.57e4i)T2 1 + (-1.48e4 - 2.57e4i)T^{2}
37 1+5.06e4T2 1 + 5.06e4T^{2}
41 1+(−148.−177.i)T+(−1.19e4+6.78e4i)T2 1 + (-148. - 177. i)T + (-1.19e4 + 6.78e4i)T^{2}
43 1+(529.+192.i)T+(6.09e4+5.11e4i)T2 1 + (529. + 192. i)T + (6.09e4 + 5.11e4i)T^{2}
47 1+(9.75e4+3.55e4i)T2 1 + (9.75e4 + 3.55e4i)T^{2}
53 1+(1.14e5−9.56e4i)T2 1 + (1.14e5 - 9.56e4i)T^{2}
59 1+(−780.+137.i)T+(1.92e5−7.02e4i)T2 1 + (-780. + 137. i)T + (1.92e5 - 7.02e4i)T^{2}
61 1+(−1.73e5+1.45e5i)T2 1 + (-1.73e5 + 1.45e5i)T^{2}
67 1+(−989.−174.i)T+(2.82e5+1.02e5i)T2 1 + (-989. - 174. i)T + (2.82e5 + 1.02e5i)T^{2}
71 1+(2.74e5+2.30e5i)T2 1 + (2.74e5 + 2.30e5i)T^{2}
73 1+(−828.+695.i)T+(6.75e4−3.83e5i)T2 1 + (-828. + 695. i)T + (6.75e4 - 3.83e5i)T^{2}
79 1+(8.56e4−4.85e5i)T2 1 + (8.56e4 - 4.85e5i)T^{2}
83 1+(750.−1.30e3i)T+(−2.85e5−4.95e5i)T2 1 + (750. - 1.30e3i)T + (-2.85e5 - 4.95e5i)T^{2}
89 1+(−998.+1.18e3i)T+(−1.22e5−6.94e5i)T2 1 + (-998. + 1.18e3i)T + (-1.22e5 - 6.94e5i)T^{2}
97 1+(−690.+121.i)T+(8.57e5−3.12e5i)T2 1 + (-690. + 121. i)T + (8.57e5 - 3.12e5i)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

−12.96223477658494971783905144126, −11.39979544961405474650181398290, −10.41011045896599701014566174721, −9.728457385777116695349208429615, −8.852521234056794221984483123096, −8.138242389345972128736139124804, −6.78789371008576191925981448444, −4.99652956535270464342227418017, −3.57859315060834975182625151706, −2.13492000025554616971096775820, 0.856748576121686104493027374618, 2.22075539093337598479561376346, 3.39619935824342537616619862505, 6.27447333148520681368330301809, 7.08377941808247979339685443593, 8.198123186365362086699420802683, 8.700661126753645737556492718148, 9.793807277229731576281343492364, 11.22892870130606601609593593685, 12.22734625895099001876013048595

Graph of the ZZ-function along the critical line