↑

Properties

Label 2-5e4-25.6-c1-0-4
Degree 22
Conductor 625625
Sign 0.998−0.0627i0.998 - 0.0627i
Analytic cond. 4.990654.99065
Root an. cond. 2.233972.23397
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  + (−0.713 − 2.19i)2-s + (0.384 − 0.279i)3-s + (−2.69 + 1.95i)4-s + (−0.887 − 0.644i)6-s − 3.03·7-s + (2.48 + 1.80i)8-s + (−0.857 + 2.63i)9-s + (0.618 + 1.90i)11-s + (−0.489 + 1.50i)12-s + (0.441 − 1.35i)13-s + (2.16 + 6.66i)14-s + (0.136 − 0.420i)16-s + (1.50 + 1.09i)17-s + 6.40·18-s + (−0.730 − 0.530i)19-s + ⋯
L(s)  = 1  + (−0.504 − 1.55i)2-s + (0.221 − 0.161i)3-s + (−1.34 + 0.979i)4-s + (−0.362 − 0.263i)6-s − 1.14·7-s + (0.880 + 0.639i)8-s + (−0.285 + 0.879i)9-s + (0.186 + 0.573i)11-s + (−0.141 + 0.434i)12-s + (0.122 − 0.376i)13-s + (0.578 + 1.78i)14-s + (0.0341 − 0.105i)16-s + (0.365 + 0.265i)17-s + 1.51·18-s + (−0.167 − 0.121i)19-s + ⋯

Functional equation

Λ(s)=(625s/2 ΓC(s) L(s)=((0.998−0.0627i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(625s/2 ΓC(s+1/2) L(s)=((0.998−0.0627i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 625625    =    545^{4}
Sign: 0.998−0.0627i0.998 - 0.0627i
Analytic conductor: 4.990654.99065
Root analytic conductor: 2.233972.23397
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ625(126,⋅)\chi_{625} (126, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 625, ( :1/2), 0.998−0.0627i)(2,\ 625,\ (\ :1/2),\ 0.998 - 0.0627i)

Particular Values

L(1)L(1) ≈\approx 0.596338+0.0187407i0.596338 + 0.0187407i
L(12)L(\frac12) ≈\approx 0.596338+0.0187407i0.596338 + 0.0187407i
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)
bad5 1 1
good2 1+(0.713+2.19i)T+(−1.61+1.17i)T2 1 + (0.713 + 2.19i)T + (-1.61 + 1.17i)T^{2}
3 1+(−0.384+0.279i)T+(0.927−2.85i)T2 1 + (-0.384 + 0.279i)T + (0.927 - 2.85i)T^{2}
7 1+3.03T+7T2 1 + 3.03T + 7T^{2}
11 1+(−0.618−1.90i)T+(−8.89+6.46i)T2 1 + (-0.618 - 1.90i)T + (-8.89 + 6.46i)T^{2}
13 1+(−0.441+1.35i)T+(−10.5−7.64i)T2 1 + (-0.441 + 1.35i)T + (-10.5 - 7.64i)T^{2}
17 1+(−1.50−1.09i)T+(5.25+16.1i)T2 1 + (-1.50 - 1.09i)T + (5.25 + 16.1i)T^{2}
19 1+(0.730+0.530i)T+(5.87+18.0i)T2 1 + (0.730 + 0.530i)T + (5.87 + 18.0i)T^{2}
23 1+(−1.02−3.16i)T+(−18.6+13.5i)T2 1 + (-1.02 - 3.16i)T + (-18.6 + 13.5i)T^{2}
29 1+(3.20−2.32i)T+(8.96−27.5i)T2 1 + (3.20 - 2.32i)T + (8.96 - 27.5i)T^{2}
31 1+(−5.21−3.78i)T+(9.57+29.4i)T2 1 + (-5.21 - 3.78i)T + (9.57 + 29.4i)T^{2}
37 1+(1.18−3.63i)T+(−29.9−21.7i)T2 1 + (1.18 - 3.63i)T + (-29.9 - 21.7i)T^{2}
41 1+(0.566−1.74i)T+(−33.1−24.0i)T2 1 + (0.566 - 1.74i)T + (-33.1 - 24.0i)T^{2}
43 1−3.59T+43T2 1 - 3.59T + 43T^{2}
47 1+(3.88−2.82i)T+(14.5−44.6i)T2 1 + (3.88 - 2.82i)T + (14.5 - 44.6i)T^{2}
53 1+(7.68−5.58i)T+(16.3−50.4i)T2 1 + (7.68 - 5.58i)T + (16.3 - 50.4i)T^{2}
59 1+(−3.28+10.1i)T+(−47.7−34.6i)T2 1 + (-3.28 + 10.1i)T + (-47.7 - 34.6i)T^{2}
61 1+(−4.41−13.5i)T+(−49.3+35.8i)T2 1 + (-4.41 - 13.5i)T + (-49.3 + 35.8i)T^{2}
67 1+(8.64+6.28i)T+(20.7+63.7i)T2 1 + (8.64 + 6.28i)T + (20.7 + 63.7i)T^{2}
71 1+(10.0−7.32i)T+(21.9−67.5i)T2 1 + (10.0 - 7.32i)T + (21.9 - 67.5i)T^{2}
73 1+(0.0827+0.254i)T+(−59.0+42.9i)T2 1 + (0.0827 + 0.254i)T + (-59.0 + 42.9i)T^{2}
79 1+(−6.93+5.03i)T+(24.4−75.1i)T2 1 + (-6.93 + 5.03i)T + (24.4 - 75.1i)T^{2}
83 1+(10.2+7.41i)T+(25.6+78.9i)T2 1 + (10.2 + 7.41i)T + (25.6 + 78.9i)T^{2}
89 1+(1.47+4.53i)T+(−72.0+52.3i)T2 1 + (1.47 + 4.53i)T + (-72.0 + 52.3i)T^{2}
97 1+(8.05−5.85i)T+(29.9−92.2i)T2 1 + (8.05 - 5.85i)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

−10.50241224257397284840976882660, −9.943279928854983457418878919515, −9.168672300875782641572355454524, −8.335862739684507776961413404040, −7.31254261228001964311882502128, −6.10135764910405274155832086908, −4.72513616523534872158595593619, −3.44237156230297242662585876078, −2.74392902853022881341437005232, −1.48417765416622127209427782830, 0.38723235570199566519346775603, 3.00019534410358030816140109604, 4.15042437662491425967491422160, 5.59382894793147770674556083883, 6.32673647371329236526848442478, 6.85714483758297597974021921694, 7.977383952205929165976296870740, 8.810759929499990106821755110419, 9.421041306267646508548504566427, 10.05352159186660096876441794660

Graph of the ZZ-function along the critical line