↑

Properties

Label 2-20e2-25.19-c1-0-8
Degree 22
Conductor 400400
Sign 0.277+0.960i0.277 + 0.960i
Analytic cond. 3.194013.19401
Root an. cond. 1.787181.78718
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  + (−0.0350 − 0.0481i)3-s + (−0.243 − 2.22i)5-s + 1.71i·7-s + (0.925 − 2.84i)9-s + (−0.573 − 1.76i)11-s + (0.533 + 0.173i)13-s + (−0.0985 + 0.0895i)15-s + (2.29 − 3.15i)17-s + (−4.69 − 3.40i)19-s + (0.0827 − 0.0601i)21-s + (6.95 − 2.25i)23-s + (−4.88 + 1.08i)25-s + (−0.339 + 0.110i)27-s + (2.13 − 1.55i)29-s + (−1.56 − 1.13i)31-s + ⋯
L(s)  = 1  + (−0.0202 − 0.0278i)3-s + (−0.108 − 0.994i)5-s + 0.648i·7-s + (0.308 − 0.949i)9-s + (−0.172 − 0.532i)11-s + (0.148 + 0.0481i)13-s + (−0.0254 + 0.0231i)15-s + (0.556 − 0.766i)17-s + (−1.07 − 0.782i)19-s + (0.0180 − 0.0131i)21-s + (1.45 − 0.471i)23-s + (−0.976 + 0.216i)25-s + (−0.0653 + 0.0212i)27-s + (0.396 − 0.288i)29-s + (−0.280 − 0.203i)31-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((0.277+0.960i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+1/2) L(s)=((0.277+0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 0.277+0.960i0.277 + 0.960i
Analytic conductor: 3.194013.19401
Root analytic conductor: 1.787181.78718
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ400(369,⋅)\chi_{400} (369, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :1/2), 0.277+0.960i)(2,\ 400,\ (\ :1/2),\ 0.277 + 0.960i)

Particular Values

L(1)L(1) ≈\approx 1.01369−0.762288i1.01369 - 0.762288i
L(12)L(\frac12) ≈\approx 1.01369−0.762288i1.01369 - 0.762288i
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+(0.243+2.22i)T 1 + (0.243 + 2.22i)T
good3 1+(0.0350+0.0481i)T+(−0.927+2.85i)T2 1 + (0.0350 + 0.0481i)T + (-0.927 + 2.85i)T^{2}
7 1−1.71iT−7T2 1 - 1.71iT - 7T^{2}
11 1+(0.573+1.76i)T+(−8.89+6.46i)T2 1 + (0.573 + 1.76i)T + (-8.89 + 6.46i)T^{2}
13 1+(−0.533−0.173i)T+(10.5+7.64i)T2 1 + (-0.533 - 0.173i)T + (10.5 + 7.64i)T^{2}
17 1+(−2.29+3.15i)T+(−5.25−16.1i)T2 1 + (-2.29 + 3.15i)T + (-5.25 - 16.1i)T^{2}
19 1+(4.69+3.40i)T+(5.87+18.0i)T2 1 + (4.69 + 3.40i)T + (5.87 + 18.0i)T^{2}
23 1+(−6.95+2.25i)T+(18.6−13.5i)T2 1 + (-6.95 + 2.25i)T + (18.6 - 13.5i)T^{2}
29 1+(−2.13+1.55i)T+(8.96−27.5i)T2 1 + (-2.13 + 1.55i)T + (8.96 - 27.5i)T^{2}
31 1+(1.56+1.13i)T+(9.57+29.4i)T2 1 + (1.56 + 1.13i)T + (9.57 + 29.4i)T^{2}
37 1+(−0.725−0.235i)T+(29.9+21.7i)T2 1 + (-0.725 - 0.235i)T + (29.9 + 21.7i)T^{2}
41 1+(3.00−9.25i)T+(−33.1−24.0i)T2 1 + (3.00 - 9.25i)T + (-33.1 - 24.0i)T^{2}
43 1+9.68iT−43T2 1 + 9.68iT - 43T^{2}
47 1+(−6.11−8.42i)T+(−14.5+44.6i)T2 1 + (-6.11 - 8.42i)T + (-14.5 + 44.6i)T^{2}
53 1+(−0.654−0.900i)T+(−16.3+50.4i)T2 1 + (-0.654 - 0.900i)T + (-16.3 + 50.4i)T^{2}
59 1+(−1.56+4.81i)T+(−47.7−34.6i)T2 1 + (-1.56 + 4.81i)T + (-47.7 - 34.6i)T^{2}
61 1+(−4.18−12.8i)T+(−49.3+35.8i)T2 1 + (-4.18 - 12.8i)T + (-49.3 + 35.8i)T^{2}
67 1+(0.733−1.01i)T+(−20.7−63.7i)T2 1 + (0.733 - 1.01i)T + (-20.7 - 63.7i)T^{2}
71 1+(10.8−7.91i)T+(21.9−67.5i)T2 1 + (10.8 - 7.91i)T + (21.9 - 67.5i)T^{2}
73 1+(3.88−1.26i)T+(59.0−42.9i)T2 1 + (3.88 - 1.26i)T + (59.0 - 42.9i)T^{2}
79 1+(5.25−3.81i)T+(24.4−75.1i)T2 1 + (5.25 - 3.81i)T + (24.4 - 75.1i)T^{2}
83 1+(0.536−0.738i)T+(−25.6−78.9i)T2 1 + (0.536 - 0.738i)T + (-25.6 - 78.9i)T^{2}
89 1+(1.61+4.97i)T+(−72.0+52.3i)T2 1 + (1.61 + 4.97i)T + (-72.0 + 52.3i)T^{2}
97 1+(−8.54−11.7i)T+(−29.9+92.2i)T2 1 + (-8.54 - 11.7i)T + (-29.9 + 92.2i)T^{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

−11.29046468864710405887082976307, −10.05746227182764487996910221565, −8.920495841606337698976053128871, −8.753183490240656630659839295270, −7.35544729636807461038544397349, −6.24075364652821014609966711322, −5.24414941200462198029986453941, −4.20838973725144079613350811569, −2.78059611543843484039724790666, −0.888189619125443349592009144819, 1.91240792561667663957398570312, 3.38224067593738820637766392539, 4.47804220377239275462041105104, 5.76618248390494914030577634694, 6.97469158315510959053635772195, 7.55508523254381904472272384624, 8.578850358210343406171883612569, 10.05493452682998733994648874932, 10.50964311990614001241461335567, 11.15782624250177507613279662292

Graph of the ZZ-function along the critical line