↑

Properties

Label 2-700-7.2-c1-0-10
Degree 22
Conductor 700700
Sign −0.989+0.142i-0.989 + 0.142i
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  + (−0.866 − 1.5i)3-s + (0.209 + 2.63i)7-s + (−1.13 − 1.97i)11-s − 6.09·13-s + (−2.38 − 4.13i)17-s + (2.13 − 3.70i)19-s + (3.77 − 2.59i)21-s + (−0.447 + 0.774i)23-s − 5.19·27-s − 3.27·29-s + (2.13 + 3.70i)31-s + (−1.97 + 3.41i)33-s + (2.80 − 4.86i)37-s + (5.27 + 9.13i)39-s − 11.2·41-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (0.0791 + 0.996i)7-s + (−0.342 − 0.594i)11-s − 1.68·13-s + (−0.579 − 1.00i)17-s + (0.490 − 0.849i)19-s + (0.823 − 0.566i)21-s + (−0.0932 + 0.161i)23-s − 1.00·27-s − 0.608·29-s + (0.383 + 0.664i)31-s + (−0.342 + 0.594i)33-s + (0.461 − 0.799i)37-s + (0.844 + 1.46i)39-s − 1.76·41-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.989+0.142i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((−0.989+0.142i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.989+0.142i-0.989 + 0.142i
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ700(401,⋅)\chi_{700} (401, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), −0.989+0.142i)(2,\ 700,\ (\ :1/2),\ -0.989 + 0.142i)

Particular Values

L(1)L(1) ≈\approx 0.0338750−0.474397i0.0338750 - 0.474397i
L(12)L(\frac12) ≈\approx 0.0338750−0.474397i0.0338750 - 0.474397i
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 1
5 1 1
7 1+(−0.209−2.63i)T 1 + (-0.209 - 2.63i)T
good3 1+(0.866+1.5i)T+(−1.5+2.59i)T2 1 + (0.866 + 1.5i)T + (-1.5 + 2.59i)T^{2}
11 1+(1.13+1.97i)T+(−5.5+9.52i)T2 1 + (1.13 + 1.97i)T + (-5.5 + 9.52i)T^{2}
13 1+6.09T+13T2 1 + 6.09T + 13T^{2}
17 1+(2.38+4.13i)T+(−8.5+14.7i)T2 1 + (2.38 + 4.13i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.13+3.70i)T+(−9.5−16.4i)T2 1 + (-2.13 + 3.70i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.447−0.774i)T+(−11.5−19.9i)T2 1 + (0.447 - 0.774i)T + (-11.5 - 19.9i)T^{2}
29 1+3.27T+29T2 1 + 3.27T + 29T^{2}
31 1+(−2.13−3.70i)T+(−15.5+26.8i)T2 1 + (-2.13 - 3.70i)T + (-15.5 + 26.8i)T^{2}
37 1+(−2.80+4.86i)T+(−18.5−32.0i)T2 1 + (-2.80 + 4.86i)T + (-18.5 - 32.0i)T^{2}
41 1+11.2T+41T2 1 + 11.2T + 41T^{2}
43 1+6.50T+43T2 1 + 6.50T + 43T^{2}
47 1+(1.07−1.86i)T+(−23.5−40.7i)T2 1 + (1.07 - 1.86i)T + (-23.5 - 40.7i)T^{2}
53 1+(3.70+6.41i)T+(−26.5+45.8i)T2 1 + (3.70 + 6.41i)T + (-26.5 + 45.8i)T^{2}
59 1+(2.13+3.70i)T+(−29.5+51.0i)T2 1 + (2.13 + 3.70i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.774−1.34i)T+(−30.5−52.8i)T2 1 + (0.774 - 1.34i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.95−12.0i)T+(−33.5+58.0i)T2 1 + (-6.95 - 12.0i)T + (-33.5 + 58.0i)T^{2}
71 1−10.5T+71T2 1 - 10.5T + 71T^{2}
73 1+(1.07+1.86i)T+(−36.5+63.2i)T2 1 + (1.07 + 1.86i)T + (-36.5 + 63.2i)T^{2}
79 1+(−0.137+0.238i)T+(−39.5−68.4i)T2 1 + (-0.137 + 0.238i)T + (-39.5 - 68.4i)T^{2}
83 1−5.67T+83T2 1 - 5.67T + 83T^{2}
89 1+(−3.5+6.06i)T+(−44.5−77.0i)T2 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2}
97 1+6.92T+97T2 1 + 6.92T + 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

−9.879969789478787647082630039523, −9.248116554998555694967777629158, −8.212097007057137199555553781953, −7.21887912982848804789734777525, −6.64528495449458619963439348184, −5.46994001724909168804647473981, −4.88897885618587086111051803425, −3.06303144955831308829779934307, −2.04912056122037259148078230545, −0.24770195952458898628909719282, 1.96944228998277801746996768337, 3.62245920364958077236720398207, 4.59633971808217535230569560533, 5.12381119562401342509828192461, 6.43230099121010075670292879776, 7.44982948776909452537465269471, 8.094472272121280566945831966690, 9.624135349851606824640857597018, 10.03288612646902664732071216852, 10.62169996536594050979740864095

Graph of the ZZ-function along the critical line