↑

Properties

Label 2-700-140.59-c1-0-10
Degree 22
Conductor 700700
Sign 0.999−0.0422i0.999 - 0.0422i
Analytic cond. 5.589525.58952
Root an. cond. 2.364212.36421
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.626 + 1.26i)2-s + (−2.59 − 1.49i)3-s + (−1.21 − 1.58i)4-s + (3.52 − 2.35i)6-s + (−1.65 + 2.06i)7-s + (2.77 − 0.546i)8-s + (2.99 + 5.18i)9-s + (−1.93 − 1.11i)11-s + (0.774 + 5.94i)12-s − 3.17·13-s + (−1.57 − 3.39i)14-s + (−1.04 + 3.86i)16-s + (−1.72 + 2.98i)17-s + (−8.45 + 0.548i)18-s + (−1.02 − 1.77i)19-s + ⋯
L(s)  = 1  + (−0.442 + 0.896i)2-s + (−1.49 − 0.865i)3-s + (−0.607 − 0.794i)4-s + (1.43 − 0.960i)6-s + (−0.627 + 0.778i)7-s + (0.981 − 0.193i)8-s + (0.998 + 1.72i)9-s + (−0.584 − 0.337i)11-s + (0.223 + 1.71i)12-s − 0.879·13-s + (−0.420 − 0.907i)14-s + (−0.261 + 0.965i)16-s + (−0.417 + 0.723i)17-s + (−1.99 + 0.129i)18-s + (−0.235 − 0.407i)19-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((0.999−0.0422i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0422i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((0.999−0.0422i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0422i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 0.999−0.0422i0.999 - 0.0422i
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ700(199,⋅)\chi_{700} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), 0.999−0.0422i)(2,\ 700,\ (\ :1/2),\ 0.999 - 0.0422i)

Particular Values

L(1)L(1) ≈\approx 0.415286+0.00877363i0.415286 + 0.00877363i
L(12)L(\frac12) ≈\approx 0.415286+0.00877363i0.415286 + 0.00877363i
L(32)L(\frac{3}{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+(0.626−1.26i)T 1 + (0.626 - 1.26i)T
5 1 1
7 1+(1.65−2.06i)T 1 + (1.65 - 2.06i)T
good3 1+(2.59+1.49i)T+(1.5+2.59i)T2 1 + (2.59 + 1.49i)T + (1.5 + 2.59i)T^{2}
11 1+(1.93+1.11i)T+(5.5+9.52i)T2 1 + (1.93 + 1.11i)T + (5.5 + 9.52i)T^{2}
13 1+3.17T+13T2 1 + 3.17T + 13T^{2}
17 1+(1.72−2.98i)T+(−8.5−14.7i)T2 1 + (1.72 - 2.98i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.02+1.77i)T+(−9.5+16.4i)T2 1 + (1.02 + 1.77i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.33+2.30i)T+(−11.5+19.9i)T2 1 + (1.33 + 2.30i)T + (-11.5 + 19.9i)T^{2}
29 1−7.38T+29T2 1 - 7.38T + 29T^{2}
31 1+(2.44−4.23i)T+(−15.5−26.8i)T2 1 + (2.44 - 4.23i)T + (-15.5 - 26.8i)T^{2}
37 1+(−9.69+5.59i)T+(18.5−32.0i)T2 1 + (-9.69 + 5.59i)T + (18.5 - 32.0i)T^{2}
41 1−1.46iT−41T2 1 - 1.46iT - 41T^{2}
43 1−9.95T+43T2 1 - 9.95T + 43T^{2}
47 1+(−5.30+3.06i)T+(23.5−40.7i)T2 1 + (-5.30 + 3.06i)T + (23.5 - 40.7i)T^{2}
53 1+(4.03+2.32i)T+(26.5+45.8i)T2 1 + (4.03 + 2.32i)T + (26.5 + 45.8i)T^{2}
59 1+(−3.55+6.16i)T+(−29.5−51.0i)T2 1 + (-3.55 + 6.16i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.19−1.26i)T+(30.5−52.8i)T2 1 + (2.19 - 1.26i)T + (30.5 - 52.8i)T^{2}
67 1+(0.0263−0.0456i)T+(−33.5−58.0i)T2 1 + (0.0263 - 0.0456i)T + (-33.5 - 58.0i)T^{2}
71 1+0.212iT−71T2 1 + 0.212iT - 71T^{2}
73 1+(−7.43+12.8i)T+(−36.5−63.2i)T2 1 + (-7.43 + 12.8i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.399−0.230i)T+(39.5−68.4i)T2 1 + (0.399 - 0.230i)T + (39.5 - 68.4i)T^{2}
83 1−10.9iT−83T2 1 - 10.9iT - 83T^{2}
89 1+(6.07−3.51i)T+(44.5−77.0i)T2 1 + (6.07 - 3.51i)T + (44.5 - 77.0i)T^{2}
97 1+0.185T+97T2 1 + 0.185T + 97T^{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

−10.53764919738803387288431796762, −9.584400655903988648186587951788, −8.531133992164955848602822345835, −7.62159417974508884694703711678, −6.73497348890372261878931692576, −6.13907019087926746625863759254, −5.44886037164685599224514295602, −4.56479923498339720794524765511, −2.28519460728039652710812137784, −0.53687160210253421575352785457, 0.68514306585376320189968664801, 2.74541968016351454133374620769, 4.16560738652823021347041127412, 4.64520098991671369582230835574, 5.78640755082195305105493754888, 6.94720939401191949737417110289, 7.81167706398649020387945560564, 9.359709516807034578981922251015, 9.841788488788571417293536812449, 10.45935036050292060910411530143

Graph of the ZZ-function along the critical line