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Properties

Label 2-624-52.31-c3-0-37
Degree 22
Conductor 624624
Sign 0.318+0.947i0.318 + 0.947i
Analytic cond. 36.817136.8171
Root an. cond. 6.067716.06771
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  + 3i·3-s + (7.87 − 7.87i)5-s + (15.6 − 15.6i)7-s − 9·9-s + (35.2 − 35.2i)11-s + (39.8 − 24.6i)13-s + (23.6 + 23.6i)15-s − 8.37i·17-s + (−3.94 − 3.94i)19-s + (46.8 + 46.8i)21-s − 192.·23-s + 0.965i·25-s − 27i·27-s − 8.58·29-s + (−214. − 214. i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.704 − 0.704i)5-s + (0.843 − 0.843i)7-s − 0.333·9-s + (0.965 − 0.965i)11-s + (0.850 − 0.526i)13-s + (0.406 + 0.406i)15-s − 0.119i·17-s + (−0.0475 − 0.0475i)19-s + (0.487 + 0.487i)21-s − 1.74·23-s + 0.00772i·25-s − 0.192i·27-s − 0.0549·29-s + (−1.24 − 1.24i)31-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=((0.318+0.947i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+3/2) L(s)=((0.318+0.947i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: 0.318+0.947i0.318 + 0.947i
Analytic conductor: 36.817136.8171
Root analytic conductor: 6.067716.06771
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ624(31,⋅)\chi_{624} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 624, ( :3/2), 0.318+0.947i)(2,\ 624,\ (\ :3/2),\ 0.318 + 0.947i)

Particular Values

L(2)L(2) ≈\approx 2.4646754132.464675413
L(12)L(\frac12) ≈\approx 2.4646754132.464675413
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 1−3iT 1 - 3iT
13 1+(−39.8+24.6i)T 1 + (-39.8 + 24.6i)T
good5 1+(−7.87+7.87i)T−125iT2 1 + (-7.87 + 7.87i)T - 125iT^{2}
7 1+(−15.6+15.6i)T−343iT2 1 + (-15.6 + 15.6i)T - 343iT^{2}
11 1+(−35.2+35.2i)T−1.33e3iT2 1 + (-35.2 + 35.2i)T - 1.33e3iT^{2}
17 1+8.37iT−4.91e3T2 1 + 8.37iT - 4.91e3T^{2}
19 1+(3.94+3.94i)T+6.85e3iT2 1 + (3.94 + 3.94i)T + 6.85e3iT^{2}
23 1+192.T+1.21e4T2 1 + 192.T + 1.21e4T^{2}
29 1+8.58T+2.43e4T2 1 + 8.58T + 2.43e4T^{2}
31 1+(214.+214.i)T+2.97e4iT2 1 + (214. + 214. i)T + 2.97e4iT^{2}
37 1+(86.7+86.7i)T+5.06e4iT2 1 + (86.7 + 86.7i)T + 5.06e4iT^{2}
41 1+(−210.+210.i)T−6.89e4iT2 1 + (-210. + 210. i)T - 6.89e4iT^{2}
43 1+23.0T+7.95e4T2 1 + 23.0T + 7.95e4T^{2}
47 1+(201.−201.i)T−1.03e5iT2 1 + (201. - 201. i)T - 1.03e5iT^{2}
53 1−73.3T+1.48e5T2 1 - 73.3T + 1.48e5T^{2}
59 1+(240.−240.i)T−2.05e5iT2 1 + (240. - 240. i)T - 2.05e5iT^{2}
61 1+244.T+2.26e5T2 1 + 244.T + 2.26e5T^{2}
67 1+(−579.−579.i)T+3.00e5iT2 1 + (-579. - 579. i)T + 3.00e5iT^{2}
71 1+(−15.1−15.1i)T+3.57e5iT2 1 + (-15.1 - 15.1i)T + 3.57e5iT^{2}
73 1+(−564.−564.i)T+3.89e5iT2 1 + (-564. - 564. i)T + 3.89e5iT^{2}
79 1+507.iT−4.93e5T2 1 + 507. iT - 4.93e5T^{2}
83 1+(316.+316.i)T+5.71e5iT2 1 + (316. + 316. i)T + 5.71e5iT^{2}
89 1+(−818.−818.i)T+7.04e5iT2 1 + (-818. - 818. i)T + 7.04e5iT^{2}
97 1+(−1.16e3+1.16e3i)T−9.12e5iT2 1 + (-1.16e3 + 1.16e3i)T - 9.12e5iT^{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

−10.00215785987538787860157058404, −9.157068500924300488071836104844, −8.448344666479693887598396534661, −7.56010662115008338358008736913, −6.09065127909394839319091276359, −5.54392059653323310936959646017, −4.30014345873142999747276268523, −3.62643098060382075334988131407, −1.80657922466578258315293002602, −0.71433434350634372443198604577, 1.65338381517421663343594886386, 2.10996391767897053127233782991, 3.67328236189005273401370597405, 4.97068404970793203531783879776, 6.13789303149477211059804738951, 6.59548677962459249981839852383, 7.72616829392497531040220029266, 8.647262836880531603493320136038, 9.442797929210295860039247010727, 10.40206786912738959308233802496

Graph of the ZZ-function along the critical line