↑

Properties

Label 2-700-35.19-c4-0-29
Degree 22
Conductor 700700
Sign 0.332+0.943i0.332 + 0.943i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((0.332+0.943i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+2) L(s)=((0.332+0.943i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 0.332+0.943i0.332 + 0.943i
Analytic conductor: 72.358972.3589
Root analytic conductor: 8.506408.50640
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ700(649,⋅)\chi_{700} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :2), 0.332+0.943i)(2,\ 700,\ (\ :2),\ 0.332 + 0.943i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 2.1122612092.112261209
L(12)L(\frac12) ≈\approx 2.1122612092.112261209
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+(24.5+42.3i)T 1 + (24.5 + 42.3i)T
good3 1+(0.784+1.35i)T+(−40.5+70.1i)T2 1 + (0.784 + 1.35i)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.T+2.85e4T2 1 - 136.T + 2.85e4T^{2}
17 1+(−131.−227.i)T+(−4.17e4+7.23e4i)T2 1 + (-131. - 227. 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+(−648.−374.i)T+(1.39e5+2.42e5i)T2 1 + (-648. - 374. 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+(−645.−372.i)T+(9.37e5+1.62e6i)T2 1 + (-645. - 372. i)T + (9.37e5 + 1.62e6i)T^{2}
41 1+2.47e3iT−2.82e6T2 1 + 2.47e3iT - 2.82e6T^{2}
43 1+2.63e3iT−3.41e6T2 1 + 2.63e3iT - 3.41e6T^{2}
47 1+(334.−579.i)T+(−2.43e6−4.22e6i)T2 1 + (334. - 579. i)T + (-2.43e6 - 4.22e6i)T^{2}
53 1+(1.75e3−1.01e3i)T+(3.94e6−6.83e6i)T2 1 + (1.75e3 - 1.01e3i)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+(−6.54e3+3.77e3i)T+(1.00e7−1.74e7i)T2 1 + (-6.54e3 + 3.77e3i)T + (1.00e7 - 1.74e7i)T^{2}
71 1−7.57e3T+2.54e7T2 1 - 7.57e3T + 2.54e7T^{2}
73 1+(−1.89e3−3.28e3i)T+(−1.41e7+2.45e7i)T2 1 + (-1.89e3 - 3.28e3i)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 1−2.18e3T+4.74e7T2 1 - 2.18e3T + 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 1−9.98e3T+8.85e7T2 1 - 9.98e3T + 8.85e7T^{2}
show more
show less
   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

−9.689347806033623953510820598783, −8.981220274709314256636672436479, −7.78704622309247144140991889484, −7.07212974545706711247335417261, −6.22022921161667057831606038491, −5.27968105974971188673304065730, −3.74843669179406048002676374853, −3.45672876644226539948977370072, −1.51981169596749391062039551301, −0.64188659835651935142996673318, 1.00010464750136170943107109039, 2.41767808830725214311194952972, 3.34485570072541265723997828397, 4.75040963051655304827739766052, 5.41918981130907074703464210600, 6.49322537729246587832109328994, 7.42290920277910531947998175252, 8.322970125763891306364428778031, 9.375388265137903528482307820071, 9.803539455365964199905802129884

Graph of the ZZ-function along the critical line