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Properties

Label 2-504-63.16-c1-0-18
Degree 22
Conductor 504504
Sign 0.950−0.311i0.950 - 0.311i
Analytic cond. 4.024464.02446
Root an. cond. 2.006102.00610
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 + 1.09i)3-s + 2.66·5-s + (1.94 − 1.79i)7-s + (0.613 + 2.93i)9-s + 1.36·11-s + (−2.75 − 4.77i)13-s + (3.58 + 2.91i)15-s + (−1.23 − 2.14i)17-s + (−2.19 + 3.80i)19-s + (4.57 − 0.286i)21-s − 4.69·23-s + 2.11·25-s + (−2.38 + 4.61i)27-s + (2.94 − 5.10i)29-s + (−1.55 + 2.69i)31-s + ⋯
L(s)  = 1  + (0.776 + 0.630i)3-s + 1.19·5-s + (0.735 − 0.678i)7-s + (0.204 + 0.978i)9-s + 0.411·11-s + (−0.764 − 1.32i)13-s + (0.925 + 0.752i)15-s + (−0.300 − 0.520i)17-s + (−0.503 + 0.872i)19-s + (0.998 − 0.0626i)21-s − 0.977·23-s + 0.423·25-s + (−0.458 + 0.888i)27-s + (0.547 − 0.948i)29-s + (−0.279 + 0.484i)31-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=((0.950−0.311i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+1/2) L(s)=((0.950−0.311i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: 0.950−0.311i0.950 - 0.311i
Analytic conductor: 4.024464.02446
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ504(457,⋅)\chi_{504} (457, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 504, ( :1/2), 0.950−0.311i)(2,\ 504,\ (\ :1/2),\ 0.950 - 0.311i)

Particular Values

L(1)L(1) ≈\approx 2.24491+0.357978i2.24491 + 0.357978i
L(12)L(\frac12) ≈\approx 2.24491+0.357978i2.24491 + 0.357978i
L(32)L(\frac{3}{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+(−1.34−1.09i)T 1 + (-1.34 - 1.09i)T
7 1+(−1.94+1.79i)T 1 + (-1.94 + 1.79i)T
good5 1−2.66T+5T2 1 - 2.66T + 5T^{2}
11 1−1.36T+11T2 1 - 1.36T + 11T^{2}
13 1+(2.75+4.77i)T+(−6.5+11.2i)T2 1 + (2.75 + 4.77i)T + (-6.5 + 11.2i)T^{2}
17 1+(1.23+2.14i)T+(−8.5+14.7i)T2 1 + (1.23 + 2.14i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.19−3.80i)T+(−9.5−16.4i)T2 1 + (2.19 - 3.80i)T + (-9.5 - 16.4i)T^{2}
23 1+4.69T+23T2 1 + 4.69T + 23T^{2}
29 1+(−2.94+5.10i)T+(−14.5−25.1i)T2 1 + (-2.94 + 5.10i)T + (-14.5 - 25.1i)T^{2}
31 1+(1.55−2.69i)T+(−15.5−26.8i)T2 1 + (1.55 - 2.69i)T + (-15.5 - 26.8i)T^{2}
37 1+(3.15−5.46i)T+(−18.5−32.0i)T2 1 + (3.15 - 5.46i)T + (-18.5 - 32.0i)T^{2}
41 1+(−1.38−2.40i)T+(−20.5+35.5i)T2 1 + (-1.38 - 2.40i)T + (-20.5 + 35.5i)T^{2}
43 1+(4.87−8.45i)T+(−21.5−37.2i)T2 1 + (4.87 - 8.45i)T + (-21.5 - 37.2i)T^{2}
47 1+(−5.02−8.70i)T+(−23.5+40.7i)T2 1 + (-5.02 - 8.70i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.47+2.56i)T+(−26.5+45.8i)T2 1 + (1.47 + 2.56i)T + (-26.5 + 45.8i)T^{2}
59 1+(1.77−3.07i)T+(−29.5−51.0i)T2 1 + (1.77 - 3.07i)T + (-29.5 - 51.0i)T^{2}
61 1+(0.663+1.14i)T+(−30.5+52.8i)T2 1 + (0.663 + 1.14i)T + (-30.5 + 52.8i)T^{2}
67 1+(4.14−7.18i)T+(−33.5−58.0i)T2 1 + (4.14 - 7.18i)T + (-33.5 - 58.0i)T^{2}
71 1−12.3T+71T2 1 - 12.3T + 71T^{2}
73 1+(1.11+1.93i)T+(−36.5+63.2i)T2 1 + (1.11 + 1.93i)T + (-36.5 + 63.2i)T^{2}
79 1+(6.41+11.1i)T+(−39.5+68.4i)T2 1 + (6.41 + 11.1i)T + (-39.5 + 68.4i)T^{2}
83 1+(−5.15+8.93i)T+(−41.5−71.8i)T2 1 + (-5.15 + 8.93i)T + (-41.5 - 71.8i)T^{2}
89 1+(−7.73+13.3i)T+(−44.5−77.0i)T2 1 + (-7.73 + 13.3i)T + (-44.5 - 77.0i)T^{2}
97 1+(2.55−4.42i)T+(−48.5−84.0i)T2 1 + (2.55 - 4.42i)T + (-48.5 - 84.0i)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

−10.48442793011327081403196651292, −10.14391113571537606503064241448, −9.378193844826833363830498542199, −8.239074031579735282742921773160, −7.63975082872597650649431826237, −6.24387651932524711418129934027, −5.16563262480047653374311851796, −4.27590374440593461756360694992, −2.90860496116361990194103105867, −1.73310566367429714672396927767, 1.87072224065427162314867571572, 2.27830417728386516664987463868, 4.05345465673540611656840113463, 5.33174851696915635080121793371, 6.41621224607964763127835588882, 7.12356393640237478194012699130, 8.412933034262014333977045802831, 9.053952270951411789681499316563, 9.669276966046941022905315333056, 10.86189169219777165767746513209

Graph of the ZZ-function along the critical line