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Properties

Label 2-648-9.4-c3-0-27
Degree 22
Conductor 648648
Sign 0.173+0.984i0.173 + 0.984i
Analytic cond. 38.233238.2332
Root an. cond. 6.183306.18330
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  + (7 − 12.1i)5-s + (12 + 20.7i)7-s + (−14 − 24.2i)11-s + (37 − 64.0i)13-s − 82·17-s + 92·19-s + (4 − 6.92i)23-s + (−35.5 − 61.4i)25-s + (−69 − 119. i)29-s + (−40 + 69.2i)31-s + 336·35-s + 30·37-s + (141 − 244. i)41-s + (−2 − 3.46i)43-s + (120 + 207. i)47-s + ⋯
L(s)  = 1  + (0.626 − 1.08i)5-s + (0.647 + 1.12i)7-s + (−0.383 − 0.664i)11-s + (0.789 − 1.36i)13-s − 1.16·17-s + 1.11·19-s + (0.0362 − 0.0628i)23-s + (−0.284 − 0.491i)25-s + (−0.441 − 0.765i)29-s + (−0.231 + 0.401i)31-s + 1.62·35-s + 0.133·37-s + (0.537 − 0.930i)41-s + (−0.00709 − 0.0122i)43-s + (0.372 + 0.645i)47-s + ⋯

Functional equation

Λ(s)=(648s/2 ΓC(s) L(s)=((0.173+0.984i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(648s/2 ΓC(s+3/2) L(s)=((0.173+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 648648    =    23⋅342^{3} \cdot 3^{4}
Sign: 0.173+0.984i0.173 + 0.984i
Analytic conductor: 38.233238.2332
Root analytic conductor: 6.183306.18330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ648(433,⋅)\chi_{648} (433, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 648, ( :3/2), 0.173+0.984i)(2,\ 648,\ (\ :3/2),\ 0.173 + 0.984i)

Particular Values

L(2)L(2) ≈\approx 2.2423901692.242390169
L(12)L(\frac12) ≈\approx 2.2423901692.242390169
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 1
3 1 1
good5 1+(−7+12.1i)T+(−62.5−108.i)T2 1 + (-7 + 12.1i)T + (-62.5 - 108. i)T^{2}
7 1+(−12−20.7i)T+(−171.5+297.i)T2 1 + (-12 - 20.7i)T + (-171.5 + 297. i)T^{2}
11 1+(14+24.2i)T+(−665.5+1.15e3i)T2 1 + (14 + 24.2i)T + (-665.5 + 1.15e3i)T^{2}
13 1+(−37+64.0i)T+(−1.09e3−1.90e3i)T2 1 + (-37 + 64.0i)T + (-1.09e3 - 1.90e3i)T^{2}
17 1+82T+4.91e3T2 1 + 82T + 4.91e3T^{2}
19 1−92T+6.85e3T2 1 - 92T + 6.85e3T^{2}
23 1+(−4+6.92i)T+(−6.08e3−1.05e4i)T2 1 + (-4 + 6.92i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(69+119.i)T+(−1.21e4+2.11e4i)T2 1 + (69 + 119. i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+(40−69.2i)T+(−1.48e4−2.57e4i)T2 1 + (40 - 69.2i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1−30T+5.06e4T2 1 - 30T + 5.06e4T^{2}
41 1+(−141+244.i)T+(−3.44e4−5.96e4i)T2 1 + (-141 + 244. i)T + (-3.44e4 - 5.96e4i)T^{2}
43 1+(2+3.46i)T+(−3.97e4+6.88e4i)T2 1 + (2 + 3.46i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1+(−120−207.i)T+(−5.19e4+8.99e4i)T2 1 + (-120 - 207. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1−130T+1.48e5T2 1 - 130T + 1.48e5T^{2}
59 1+(−298+516.i)T+(−1.02e5−1.77e5i)T2 1 + (-298 + 516. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−109−188.i)T+(−1.13e5+1.96e5i)T2 1 + (-109 - 188. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(−218+377.i)T+(−1.50e5−2.60e5i)T2 1 + (-218 + 377. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+856T+3.57e5T2 1 + 856T + 3.57e5T^{2}
73 1+998T+3.89e5T2 1 + 998T + 3.89e5T^{2}
79 1+(−16−27.7i)T+(−2.46e5+4.26e5i)T2 1 + (-16 - 27.7i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+(754+1.30e3i)T+(−2.85e5+4.95e5i)T2 1 + (754 + 1.30e3i)T + (-2.85e5 + 4.95e5i)T^{2}
89 1−246T+7.04e5T2 1 - 246T + 7.04e5T^{2}
97 1+(433+749.i)T+(−4.56e5+7.90e5i)T2 1 + (433 + 749. i)T + (-4.56e5 + 7.90e5i)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

−9.844233880186278535647137565263, −8.755379672257053794771258607995, −8.621405379032568533200192710895, −7.52328164065774003601834915746, −5.81504936933466900162022931619, −5.63568407027808041708922415239, −4.63184982051734542147706694343, −3.09802766053267128982406152202, −1.90189234642562492371828503222, −0.66289578997071255824265973458, 1.37942192977206193334380248314, 2.44480035873132897334945559695, 3.83017465570859733431992307674, 4.69510224292460900333135538593, 6.00217826924416378270353643239, 7.02401065066841290790610169606, 7.32910167601967850554579087301, 8.680146680326473530812861359555, 9.640955368111765398935526997859, 10.42673956495899270700617516249

Graph of the ZZ-function along the critical line