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Properties

Label 2-624-52.31-c3-0-2
Degree 22
Conductor 624624
Sign −0.916+0.399i-0.916 + 0.399i
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 + (−2.84 + 2.84i)5-s + (9.35 − 9.35i)7-s − 9·9-s + (−20.3 + 20.3i)11-s + (18.5 − 43.0i)13-s + (−8.53 − 8.53i)15-s + 31.4i·17-s + (32.5 + 32.5i)19-s + (28.0 + 28.0i)21-s − 107.·23-s + 108. i·25-s − 27i·27-s − 231.·29-s + (−3.20 − 3.20i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.254 + 0.254i)5-s + (0.505 − 0.505i)7-s − 0.333·9-s + (−0.557 + 0.557i)11-s + (0.395 − 0.918i)13-s + (−0.146 − 0.146i)15-s + 0.448i·17-s + (0.393 + 0.393i)19-s + (0.291 + 0.291i)21-s − 0.972·23-s + 0.870i·25-s − 0.192i·27-s − 1.48·29-s + (−0.0185 − 0.0185i)31-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=((−0.916+0.399i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+3/2) L(s)=((−0.916+0.399i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: −0.916+0.399i-0.916 + 0.399i
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), −0.916+0.399i)(2,\ 624,\ (\ :3/2),\ -0.916 + 0.399i)

Particular Values

L(2)L(2) ≈\approx 0.14636970740.1463697074
L(12)L(\frac12) ≈\approx 0.14636970740.1463697074
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+(−18.5+43.0i)T 1 + (-18.5 + 43.0i)T
good5 1+(2.84−2.84i)T−125iT2 1 + (2.84 - 2.84i)T - 125iT^{2}
7 1+(−9.35+9.35i)T−343iT2 1 + (-9.35 + 9.35i)T - 343iT^{2}
11 1+(20.3−20.3i)T−1.33e3iT2 1 + (20.3 - 20.3i)T - 1.33e3iT^{2}
17 1−31.4iT−4.91e3T2 1 - 31.4iT - 4.91e3T^{2}
19 1+(−32.5−32.5i)T+6.85e3iT2 1 + (-32.5 - 32.5i)T + 6.85e3iT^{2}
23 1+107.T+1.21e4T2 1 + 107.T + 1.21e4T^{2}
29 1+231.T+2.43e4T2 1 + 231.T + 2.43e4T^{2}
31 1+(3.20+3.20i)T+2.97e4iT2 1 + (3.20 + 3.20i)T + 2.97e4iT^{2}
37 1+(159.+159.i)T+5.06e4iT2 1 + (159. + 159. i)T + 5.06e4iT^{2}
41 1+(68.9−68.9i)T−6.89e4iT2 1 + (68.9 - 68.9i)T - 6.89e4iT^{2}
43 1−213.T+7.95e4T2 1 - 213.T + 7.95e4T^{2}
47 1+(−95.2+95.2i)T−1.03e5iT2 1 + (-95.2 + 95.2i)T - 1.03e5iT^{2}
53 1+647.T+1.48e5T2 1 + 647.T + 1.48e5T^{2}
59 1+(−182.+182.i)T−2.05e5iT2 1 + (-182. + 182. i)T - 2.05e5iT^{2}
61 1+263.T+2.26e5T2 1 + 263.T + 2.26e5T^{2}
67 1+(637.+637.i)T+3.00e5iT2 1 + (637. + 637. i)T + 3.00e5iT^{2}
71 1+(−229.−229.i)T+3.57e5iT2 1 + (-229. - 229. i)T + 3.57e5iT^{2}
73 1+(291.+291.i)T+3.89e5iT2 1 + (291. + 291. i)T + 3.89e5iT^{2}
79 1+633.iT−4.93e5T2 1 + 633. iT - 4.93e5T^{2}
83 1+(583.+583.i)T+5.71e5iT2 1 + (583. + 583. i)T + 5.71e5iT^{2}
89 1+(335.+335.i)T+7.04e5iT2 1 + (335. + 335. i)T + 7.04e5iT^{2}
97 1+(955.−955.i)T−9.12e5iT2 1 + (955. - 955. i)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

−10.69166041149228505161898961232, −9.994187443344325046073894934120, −9.033533035082500665752149560629, −7.85132521298178111379987937424, −7.50586311697702067328760202728, −6.02322406974307723929280042783, −5.18286151052205339116734856510, −4.08999453115787299318395740177, −3.24008607614061261745469378812, −1.71021098549038634445095940808, 0.04072425759240970652590719258, 1.53803175676026674496879003618, 2.67238974088723301315570005347, 4.04520813744437430156445977840, 5.20050610450338252909000194665, 6.05363630296037609913238841804, 7.11413304444942205573083547022, 8.030966152266213873097158544911, 8.683451796674401882361411617211, 9.557764736597928595602571977581

Graph of the ZZ-function along the critical line