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Properties

Label 2-624-52.47-c3-0-41
Degree 22
Conductor 624624
Sign 0.378−0.925i0.378 - 0.925i
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 + (−6.10 − 6.10i)5-s + (−23.8 − 23.8i)7-s − 9·9-s + (−33.3 − 33.3i)11-s + (−42.6 + 19.5i)13-s + (−18.3 + 18.3i)15-s + 20.6i·17-s + (−50.6 + 50.6i)19-s + (−71.5 + 71.5i)21-s + 38.2·23-s − 50.4i·25-s + 27i·27-s + 184.·29-s + (124. − 124. i)31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.546 − 0.546i)5-s + (−1.28 − 1.28i)7-s − 0.333·9-s + (−0.914 − 0.914i)11-s + (−0.909 + 0.416i)13-s + (−0.315 + 0.315i)15-s + 0.294i·17-s + (−0.611 + 0.611i)19-s + (−0.743 + 0.743i)21-s + 0.346·23-s − 0.403i·25-s + 0.192i·27-s + 1.17·29-s + (0.719 − 0.719i)31-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=((0.378−0.925i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+3/2) L(s)=((0.378−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: 0.378−0.925i0.378 - 0.925i
Analytic conductor: 36.817136.8171
Root analytic conductor: 6.067716.06771
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ624(463,⋅)\chi_{624} (463, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 624, ( :3/2), 0.378−0.925i)(2,\ 624,\ (\ :3/2),\ 0.378 - 0.925i)

Particular Values

L(2)L(2) ≈\approx 0.059514273880.05951427388
L(12)L(\frac12) ≈\approx 0.059514273880.05951427388
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+(42.6−19.5i)T 1 + (42.6 - 19.5i)T
good5 1+(6.10+6.10i)T+125iT2 1 + (6.10 + 6.10i)T + 125iT^{2}
7 1+(23.8+23.8i)T+343iT2 1 + (23.8 + 23.8i)T + 343iT^{2}
11 1+(33.3+33.3i)T+1.33e3iT2 1 + (33.3 + 33.3i)T + 1.33e3iT^{2}
17 1−20.6iT−4.91e3T2 1 - 20.6iT - 4.91e3T^{2}
19 1+(50.6−50.6i)T−6.85e3iT2 1 + (50.6 - 50.6i)T - 6.85e3iT^{2}
23 1−38.2T+1.21e4T2 1 - 38.2T + 1.21e4T^{2}
29 1−184.T+2.43e4T2 1 - 184.T + 2.43e4T^{2}
31 1+(−124.+124.i)T−2.97e4iT2 1 + (-124. + 124. i)T - 2.97e4iT^{2}
37 1+(−55.3+55.3i)T−5.06e4iT2 1 + (-55.3 + 55.3i)T - 5.06e4iT^{2}
41 1+(−39.7−39.7i)T+6.89e4iT2 1 + (-39.7 - 39.7i)T + 6.89e4iT^{2}
43 1−371.T+7.95e4T2 1 - 371.T + 7.95e4T^{2}
47 1+(416.+416.i)T+1.03e5iT2 1 + (416. + 416. i)T + 1.03e5iT^{2}
53 1−34.9T+1.48e5T2 1 - 34.9T + 1.48e5T^{2}
59 1+(54.1+54.1i)T+2.05e5iT2 1 + (54.1 + 54.1i)T + 2.05e5iT^{2}
61 1+613.T+2.26e5T2 1 + 613.T + 2.26e5T^{2}
67 1+(−694.+694.i)T−3.00e5iT2 1 + (-694. + 694. i)T - 3.00e5iT^{2}
71 1+(−515.+515.i)T−3.57e5iT2 1 + (-515. + 515. i)T - 3.57e5iT^{2}
73 1+(216.−216.i)T−3.89e5iT2 1 + (216. - 216. i)T - 3.89e5iT^{2}
79 1−143.iT−4.93e5T2 1 - 143. iT - 4.93e5T^{2}
83 1+(587.−587.i)T−5.71e5iT2 1 + (587. - 587. i)T - 5.71e5iT^{2}
89 1+(967.−967.i)T−7.04e5iT2 1 + (967. - 967. i)T - 7.04e5iT^{2}
97 1+(174.+174.i)T+9.12e5iT2 1 + (174. + 174. 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

−9.606070291707687050665383366902, −8.378592510293674916255292490698, −7.77833214254829017193754593399, −6.81779308709087353417491257854, −6.08133817246404113577576034941, −4.70427099315003174413683275427, −3.72674516913046692685294568697, −2.61972378087299070475356513807, −0.78381052898393190654013952159, −0.02393784988939507548725344958, 2.67140918096854675277765139288, 2.93633241977256398610128029491, 4.50749297134762011811524222167, 5.37308390084214518620267478332, 6.47241137304580928299969977752, 7.31447529338626543059667165719, 8.390976566827701927615349104161, 9.377847715758294775034744000764, 9.955559537244972055741013415857, 10.75796845935532050604588098446

Graph of the ZZ-function along the critical line