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Properties

Label 2-504-21.17-c5-0-0
Degree 22
Conductor 504504
Sign −0.988−0.154i-0.988 - 0.154i
Analytic cond. 80.833480.8334
Root an. cond. 8.990748.99074
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−14.9 − 25.9i)5-s + (−128. + 19.3i)7-s + (321. + 185. i)11-s + 148. i·13-s + (−717. + 1.24e3i)17-s + (2.41e3 − 1.39e3i)19-s + (232. − 134. i)23-s + (1.11e3 − 1.92e3i)25-s − 7.53e3i·29-s + (−2.68e3 − 1.54e3i)31-s + (2.42e3 + 3.03e3i)35-s + (7.52e3 + 1.30e4i)37-s + 2.24e3·41-s − 1.77e4·43-s + (−9.99e3 − 1.73e4i)47-s + ⋯
L(s)  = 1  + (−0.268 − 0.464i)5-s + (−0.988 + 0.149i)7-s + (0.801 + 0.462i)11-s + 0.242i·13-s + (−0.602 + 1.04i)17-s + (1.53 − 0.885i)19-s + (0.0915 − 0.0528i)23-s + (0.356 − 0.616i)25-s − 1.66i·29-s + (−0.501 − 0.289i)31-s + (0.334 + 0.419i)35-s + (0.903 + 1.56i)37-s + 0.208·41-s − 1.46·43-s + (−0.660 − 1.14i)47-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=((−0.988−0.154i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.154i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+5/2) L(s)=((−0.988−0.154i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: −0.988−0.154i-0.988 - 0.154i
Analytic conductor: 80.833480.8334
Root analytic conductor: 8.990748.99074
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ504(17,⋅)\chi_{504} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 504, ( :5/2), −0.988−0.154i)(2,\ 504,\ (\ :5/2),\ -0.988 - 0.154i)

Particular Values

L(3)L(3) ≈\approx 0.019391301980.01939130198
L(12)L(\frac12) ≈\approx 0.019391301980.01939130198
L(72)L(\frac{7}{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
7 1+(128.−19.3i)T 1 + (128. - 19.3i)T
good5 1+(14.9+25.9i)T+(−1.56e3+2.70e3i)T2 1 + (14.9 + 25.9i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(−321.−185.i)T+(8.05e4+1.39e5i)T2 1 + (-321. - 185. i)T + (8.05e4 + 1.39e5i)T^{2}
13 1−148.iT−3.71e5T2 1 - 148. iT - 3.71e5T^{2}
17 1+(717.−1.24e3i)T+(−7.09e5−1.22e6i)T2 1 + (717. - 1.24e3i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(−2.41e3+1.39e3i)T+(1.23e6−2.14e6i)T2 1 + (-2.41e3 + 1.39e3i)T + (1.23e6 - 2.14e6i)T^{2}
23 1+(−232.+134.i)T+(3.21e6−5.57e6i)T2 1 + (-232. + 134. i)T + (3.21e6 - 5.57e6i)T^{2}
29 1+7.53e3iT−2.05e7T2 1 + 7.53e3iT - 2.05e7T^{2}
31 1+(2.68e3+1.54e3i)T+(1.43e7+2.47e7i)T2 1 + (2.68e3 + 1.54e3i)T + (1.43e7 + 2.47e7i)T^{2}
37 1+(−7.52e3−1.30e4i)T+(−3.46e7+6.00e7i)T2 1 + (-7.52e3 - 1.30e4i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1−2.24e3T+1.15e8T2 1 - 2.24e3T + 1.15e8T^{2}
43 1+1.77e4T+1.47e8T2 1 + 1.77e4T + 1.47e8T^{2}
47 1+(9.99e3+1.73e4i)T+(−1.14e8+1.98e8i)T2 1 + (9.99e3 + 1.73e4i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(4.17e3+2.40e3i)T+(2.09e8+3.62e8i)T2 1 + (4.17e3 + 2.40e3i)T + (2.09e8 + 3.62e8i)T^{2}
59 1+(2.08e4−3.61e4i)T+(−3.57e8−6.19e8i)T2 1 + (2.08e4 - 3.61e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(4.30e4−2.48e4i)T+(4.22e8−7.31e8i)T2 1 + (4.30e4 - 2.48e4i)T + (4.22e8 - 7.31e8i)T^{2}
67 1+(2.52e4−4.38e4i)T+(−6.75e8−1.16e9i)T2 1 + (2.52e4 - 4.38e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1+2.94e4iT−1.80e9T2 1 + 2.94e4iT - 1.80e9T^{2}
73 1+(4.03e4+2.33e4i)T+(1.03e9+1.79e9i)T2 1 + (4.03e4 + 2.33e4i)T + (1.03e9 + 1.79e9i)T^{2}
79 1+(−7.86e3−1.36e4i)T+(−1.53e9+2.66e9i)T2 1 + (-7.86e3 - 1.36e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1−9.38e4T+3.93e9T2 1 - 9.38e4T + 3.93e9T^{2}
89 1+(−1.69e4−2.93e4i)T+(−2.79e9+4.83e9i)T2 1 + (-1.69e4 - 2.93e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1−1.15e5iT−8.58e9T2 1 - 1.15e5iT - 8.58e9T^{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

−10.39551494818781500870611552598, −9.547019210084181299553488264725, −8.936906062705961635045349232330, −7.88177575277838768269912843726, −6.78076241188385140854510509946, −6.10635605120725182661220667214, −4.77472446551225498812811337124, −3.88589534336790792065583788985, −2.71091035987673261246365360785, −1.26982414750708762750147437117, 0.00482680463135114437356507200, 1.31906061503688877521131993066, 3.08260436202328119683626019135, 3.52275811989811140744963754714, 5.00475452996236701747361588482, 6.11365840285188949359422831268, 6.98476815170403249807803904953, 7.68568456681266533743675899748, 9.125626380388914590255641228635, 9.499443216265934881263569422105

Graph of the ZZ-function along the critical line