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Properties

Label 2-675-135.4-c1-0-46
Degree 22
Conductor 675675
Sign 0.117+0.993i0.117 + 0.993i
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
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  + (0.300 − 0.826i)2-s + (1.62 − 0.592i)3-s + (0.939 + 0.788i)4-s − 1.52i·6-s + (−2.41 − 2.87i)7-s + (2.45 − 1.41i)8-s + (2.29 − 1.92i)9-s + (−0.180 − 1.02i)11-s + (1.99 + 0.726i)12-s + (−1.08 − 2.99i)13-s + (−3.10 + 1.13i)14-s + (−0.00727 − 0.0412i)16-s + (−0.405 − 0.233i)17-s + (−0.902 − 2.47i)18-s + (2.34 + 4.06i)19-s + ⋯
L(s)  = 1  + (0.212 − 0.584i)2-s + (0.939 − 0.342i)3-s + (0.469 + 0.394i)4-s − 0.621i·6-s + (−0.913 − 1.08i)7-s + (0.868 − 0.501i)8-s + (0.766 − 0.642i)9-s + (−0.0545 − 0.309i)11-s + (0.576 + 0.209i)12-s + (−0.302 − 0.830i)13-s + (−0.830 + 0.302i)14-s + (−0.00181 − 0.0103i)16-s + (−0.0982 − 0.0567i)17-s + (−0.212 − 0.584i)18-s + (0.538 + 0.932i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((0.117+0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((0.117+0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 0.117+0.993i0.117 + 0.993i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(274,⋅)\chi_{675} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), 0.117+0.993i)(2,\ 675,\ (\ :1/2),\ 0.117 + 0.993i)

Particular Values

L(1)L(1) ≈\approx 1.84900−1.64254i1.84900 - 1.64254i
L(12)L(\frac12) ≈\approx 1.84900−1.64254i1.84900 - 1.64254i
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)
bad3 1+(−1.62+0.592i)T 1 + (-1.62 + 0.592i)T
5 1 1
good2 1+(−0.300+0.826i)T+(−1.53−1.28i)T2 1 + (-0.300 + 0.826i)T + (-1.53 - 1.28i)T^{2}
7 1+(2.41+2.87i)T+(−1.21+6.89i)T2 1 + (2.41 + 2.87i)T + (-1.21 + 6.89i)T^{2}
11 1+(0.180+1.02i)T+(−10.3+3.76i)T2 1 + (0.180 + 1.02i)T + (-10.3 + 3.76i)T^{2}
13 1+(1.08+2.99i)T+(−9.95+8.35i)T2 1 + (1.08 + 2.99i)T + (-9.95 + 8.35i)T^{2}
17 1+(0.405+0.233i)T+(8.5+14.7i)T2 1 + (0.405 + 0.233i)T + (8.5 + 14.7i)T^{2}
19 1+(−2.34−4.06i)T+(−9.5+16.4i)T2 1 + (-2.34 - 4.06i)T + (-9.5 + 16.4i)T^{2}
23 1+(3.45−4.11i)T+(−3.99−22.6i)T2 1 + (3.45 - 4.11i)T + (-3.99 - 22.6i)T^{2}
29 1+(−5.45−1.98i)T+(22.2+18.6i)T2 1 + (-5.45 - 1.98i)T + (22.2 + 18.6i)T^{2}
31 1+(3.14+2.63i)T+(5.38+30.5i)T2 1 + (3.14 + 2.63i)T + (5.38 + 30.5i)T^{2}
37 1+(−3.87−2.23i)T+(18.5+32.0i)T2 1 + (-3.87 - 2.23i)T + (18.5 + 32.0i)T^{2}
41 1+(7.52−2.73i)T+(31.4−26.3i)T2 1 + (7.52 - 2.73i)T + (31.4 - 26.3i)T^{2}
43 1+(−11.9+2.11i)T+(40.4−14.7i)T2 1 + (-11.9 + 2.11i)T + (40.4 - 14.7i)T^{2}
47 1+(2.22+2.65i)T+(−8.16+46.2i)T2 1 + (2.22 + 2.65i)T + (-8.16 + 46.2i)T^{2}
53 1−8.83iT−53T2 1 - 8.83iT - 53T^{2}
59 1+(2.36−13.4i)T+(−55.4−20.1i)T2 1 + (2.36 - 13.4i)T + (-55.4 - 20.1i)T^{2}
61 1+(−7.46+6.26i)T+(10.5−60.0i)T2 1 + (-7.46 + 6.26i)T + (10.5 - 60.0i)T^{2}
67 1+(0.623+1.71i)T+(−51.3+43.0i)T2 1 + (0.623 + 1.71i)T + (-51.3 + 43.0i)T^{2}
71 1+(3.85−6.67i)T+(−35.5−61.4i)T2 1 + (3.85 - 6.67i)T + (-35.5 - 61.4i)T^{2}
73 1+(−0.705+0.407i)T+(36.5−63.2i)T2 1 + (-0.705 + 0.407i)T + (36.5 - 63.2i)T^{2}
79 1+(3.81+1.38i)T+(60.5+50.7i)T2 1 + (3.81 + 1.38i)T + (60.5 + 50.7i)T^{2}
83 1+(5.81−15.9i)T+(−63.5−53.3i)T2 1 + (5.81 - 15.9i)T + (-63.5 - 53.3i)T^{2}
89 1+(5.19+9.00i)T+(−44.5+77.0i)T2 1 + (5.19 + 9.00i)T + (-44.5 + 77.0i)T^{2}
97 1+(6.02−1.06i)T+(91.1−33.1i)T2 1 + (6.02 - 1.06i)T + (91.1 - 33.1i)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

−10.11072758669553975315984201190, −9.816878904203814502196050393020, −8.417366991611189809231459640121, −7.57071715066642209724476291327, −7.09354123535797082659809558875, −5.96869599337755112343967603580, −4.17651916823089099576722728218, −3.45827122602436038115954886676, −2.70252703236316582122298059914, −1.21192609207470900952189099355, 2.08399801724437977880282913809, 2.85449468057580580454719898934, 4.33565536775290560216198864350, 5.28307184201176531075440757803, 6.40126106107705162790474540269, 7.03917308345942727636341597833, 8.086856139536791214641369968187, 9.037621464340035479490657395462, 9.660628387202151001919311942489, 10.46600348238192142302371930962

Graph of the ZZ-function along the critical line