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Properties

Label 2-700-7.5-c4-0-33
Degree 22
Conductor 700700
Sign −0.124+0.992i-0.124 + 0.992i
Analytic cond. 72.358972.3589
Root an. cond. 8.506408.50640
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 − 0.784i)3-s + (−42.3 + 24.5i)7-s + (−39.2 + 68.0i)9-s + (−11.7 − 20.3i)11-s + 136. i·13-s + (227. − 131. i)17-s + (−387. − 223. i)19-s + (−38.3 + 66.6i)21-s + (−374. + 648. i)23-s + 250. i·27-s + 406.·29-s + (−584. + 337. i)31-s + (−31.9 − 18.4i)33-s + (372. − 645. i)37-s + (106. + 185. i)39-s + ⋯
L(s)  = 1  + (0.151 − 0.0871i)3-s + (−0.865 + 0.501i)7-s + (−0.484 + 0.839i)9-s + (−0.0971 − 0.168i)11-s + 0.806i·13-s + (0.786 − 0.454i)17-s + (−1.07 − 0.619i)19-s + (−0.0869 + 0.151i)21-s + (−0.708 + 1.22i)23-s + 0.343i·27-s + 0.483·29-s + (−0.607 + 0.350i)31-s + (−0.0293 − 0.0169i)33-s + (0.272 − 0.471i)37-s + (0.0702 + 0.121i)39-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.124+0.992i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+2) L(s)=((−0.124+0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.124+0.992i-0.124 + 0.992i
Analytic conductor: 72.358972.3589
Root analytic conductor: 8.506408.50640
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ700(201,⋅)\chi_{700} (201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :2), −0.124+0.992i)(2,\ 700,\ (\ :2),\ -0.124 + 0.992i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.63086288240.6308628824
L(12)L(\frac12) ≈\approx 0.63086288240.6308628824
L(3)L(3) 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
5 1 1
7 1+(42.3−24.5i)T 1 + (42.3 - 24.5i)T
good3 1+(−1.35+0.784i)T+(40.5−70.1i)T2 1 + (-1.35 + 0.784i)T + (40.5 - 70.1i)T^{2}
11 1+(11.7+20.3i)T+(−7.32e3+1.26e4i)T2 1 + (11.7 + 20.3i)T + (-7.32e3 + 1.26e4i)T^{2}
13 1−136.iT−2.85e4T2 1 - 136. iT - 2.85e4T^{2}
17 1+(−227.+131.i)T+(4.17e4−7.23e4i)T2 1 + (-227. + 131. i)T + (4.17e4 - 7.23e4i)T^{2}
19 1+(387.+223.i)T+(6.51e4+1.12e5i)T2 1 + (387. + 223. i)T + (6.51e4 + 1.12e5i)T^{2}
23 1+(374.−648.i)T+(−1.39e5−2.42e5i)T2 1 + (374. - 648. i)T + (-1.39e5 - 2.42e5i)T^{2}
29 1−406.T+7.07e5T2 1 - 406.T + 7.07e5T^{2}
31 1+(584.−337.i)T+(4.61e5−7.99e5i)T2 1 + (584. - 337. i)T + (4.61e5 - 7.99e5i)T^{2}
37 1+(−372.+645.i)T+(−9.37e5−1.62e6i)T2 1 + (-372. + 645. i)T + (-9.37e5 - 1.62e6i)T^{2}
41 1+2.47e3iT−2.82e6T2 1 + 2.47e3iT - 2.82e6T^{2}
43 1−2.63e3T+3.41e6T2 1 - 2.63e3T + 3.41e6T^{2}
47 1+(−579.−334.i)T+(2.43e6+4.22e6i)T2 1 + (-579. - 334. i)T + (2.43e6 + 4.22e6i)T^{2}
53 1+(1.01e3+1.75e3i)T+(−3.94e6+6.83e6i)T2 1 + (1.01e3 + 1.75e3i)T + (-3.94e6 + 6.83e6i)T^{2}
59 1+(1.01e3−585.i)T+(6.05e6−1.04e7i)T2 1 + (1.01e3 - 585. i)T + (6.05e6 - 1.04e7i)T^{2}
61 1+(2.02e3+1.16e3i)T+(6.92e6+1.19e7i)T2 1 + (2.02e3 + 1.16e3i)T + (6.92e6 + 1.19e7i)T^{2}
67 1+(3.77e3+6.54e3i)T+(−1.00e7+1.74e7i)T2 1 + (3.77e3 + 6.54e3i)T + (-1.00e7 + 1.74e7i)T^{2}
71 1−7.57e3T+2.54e7T2 1 - 7.57e3T + 2.54e7T^{2}
73 1+(3.28e3−1.89e3i)T+(1.41e7−2.45e7i)T2 1 + (3.28e3 - 1.89e3i)T + (1.41e7 - 2.45e7i)T^{2}
79 1+(3.89e3−6.75e3i)T+(−1.94e7−3.37e7i)T2 1 + (3.89e3 - 6.75e3i)T + (-1.94e7 - 3.37e7i)T^{2}
83 1−2.18e3iT−4.74e7T2 1 - 2.18e3iT - 4.74e7T^{2}
89 1+(−8.05e3−4.64e3i)T+(3.13e7+5.43e7i)T2 1 + (-8.05e3 - 4.64e3i)T + (3.13e7 + 5.43e7i)T^{2}
97 1+9.98e3iT−8.85e7T2 1 + 9.98e3iT - 8.85e7T^{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

−9.459296262633537291954463061776, −8.908966996313616291990643782074, −7.898084714242182738215987624151, −7.02581090703651825341119300307, −6.00158330215999678819000323134, −5.23629723408520391011011178030, −3.96476017225166711647526941868, −2.84677377488613590682003206551, −1.92340010697878974088420952080, −0.17080237525447218596066879154, 0.925147972311194636441000135732, 2.60618712788548907243407062803, 3.53653544930177053112784322258, 4.41285007244037864396454336851, 5.96657097696311834476979089516, 6.31218761245838807087655850024, 7.57873765781313149783827202116, 8.366096559867994967872433292203, 9.295551205739028599675098390533, 10.19347325513605957120612172365

Graph of the ZZ-function along the critical line