↑

Properties

Label 2-425-17.15-c1-0-8
Degree 22
Conductor 425425
Sign −0.865−0.500i-0.865 - 0.500i
Analytic cond. 3.393643.39364
Root an. cond. 1.842181.84218
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  + (1.44 + 1.44i)2-s + (1.22 + 2.95i)3-s + 2.18i·4-s + (−2.50 + 6.03i)6-s + (−1.08 − 0.450i)7-s + (−0.263 + 0.263i)8-s + (−5.09 + 5.09i)9-s + (1.88 − 4.54i)11-s + (−6.44 + 2.66i)12-s − 2.46i·13-s + (−0.921 − 2.22i)14-s + 3.60·16-s + (1.36 + 3.89i)17-s − 14.7·18-s + (−1.44 − 1.44i)19-s + ⋯
L(s)  = 1  + (1.02 + 1.02i)2-s + (0.706 + 1.70i)3-s + 1.09i·4-s + (−1.02 + 2.46i)6-s + (−0.411 − 0.170i)7-s + (−0.0931 + 0.0931i)8-s + (−1.69 + 1.69i)9-s + (0.567 − 1.37i)11-s + (−1.85 + 0.770i)12-s − 0.683i·13-s + (−0.246 − 0.594i)14-s + 0.900·16-s + (0.330 + 0.943i)17-s − 3.47·18-s + (−0.331 − 0.331i)19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((−0.865−0.500i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((−0.865−0.500i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −0.865−0.500i-0.865 - 0.500i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(151,⋅)\chi_{425} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), −0.865−0.500i)(2,\ 425,\ (\ :1/2),\ -0.865 - 0.500i)

Particular Values

L(1)L(1) ≈\approx 0.720693+2.68468i0.720693 + 2.68468i
L(12)L(\frac12) ≈\approx 0.720693+2.68468i0.720693 + 2.68468i
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
17 1+(−1.36−3.89i)T 1 + (-1.36 - 3.89i)T
good2 1+(−1.44−1.44i)T+2iT2 1 + (-1.44 - 1.44i)T + 2iT^{2}
3 1+(−1.22−2.95i)T+(−2.12+2.12i)T2 1 + (-1.22 - 2.95i)T + (-2.12 + 2.12i)T^{2}
7 1+(1.08+0.450i)T+(4.94+4.94i)T2 1 + (1.08 + 0.450i)T + (4.94 + 4.94i)T^{2}
11 1+(−1.88+4.54i)T+(−7.77−7.77i)T2 1 + (-1.88 + 4.54i)T + (-7.77 - 7.77i)T^{2}
13 1+2.46iT−13T2 1 + 2.46iT - 13T^{2}
19 1+(1.44+1.44i)T+19iT2 1 + (1.44 + 1.44i)T + 19iT^{2}
23 1+(−0.0455+0.109i)T+(−16.2−16.2i)T2 1 + (-0.0455 + 0.109i)T + (-16.2 - 16.2i)T^{2}
29 1+(−0.984+0.407i)T+(20.5−20.5i)T2 1 + (-0.984 + 0.407i)T + (20.5 - 20.5i)T^{2}
31 1+(−1.06−2.58i)T+(−21.9+21.9i)T2 1 + (-1.06 - 2.58i)T + (-21.9 + 21.9i)T^{2}
37 1+(0.885+2.13i)T+(−26.1+26.1i)T2 1 + (0.885 + 2.13i)T + (-26.1 + 26.1i)T^{2}
41 1+(0.662+0.274i)T+(28.9+28.9i)T2 1 + (0.662 + 0.274i)T + (28.9 + 28.9i)T^{2}
43 1+(7.13−7.13i)T−43iT2 1 + (7.13 - 7.13i)T - 43iT^{2}
47 1+3.39iT−47T2 1 + 3.39iT - 47T^{2}
53 1+(9.84+9.84i)T+53iT2 1 + (9.84 + 9.84i)T + 53iT^{2}
59 1+(−1.07+1.07i)T−59iT2 1 + (-1.07 + 1.07i)T - 59iT^{2}
61 1+(−7.46−3.09i)T+(43.1+43.1i)T2 1 + (-7.46 - 3.09i)T + (43.1 + 43.1i)T^{2}
67 1+4.92T+67T2 1 + 4.92T + 67T^{2}
71 1+(−2.53−6.13i)T+(−50.2+50.2i)T2 1 + (-2.53 - 6.13i)T + (-50.2 + 50.2i)T^{2}
73 1+(3.38−1.40i)T+(51.6−51.6i)T2 1 + (3.38 - 1.40i)T + (51.6 - 51.6i)T^{2}
79 1+(−3.13+7.55i)T+(−55.8−55.8i)T2 1 + (-3.13 + 7.55i)T + (-55.8 - 55.8i)T^{2}
83 1+(−4.30−4.30i)T+83iT2 1 + (-4.30 - 4.30i)T + 83iT^{2}
89 1−8.46iT−89T2 1 - 8.46iT - 89T^{2}
97 1+(−4.14+1.71i)T+(68.5−68.5i)T2 1 + (-4.14 + 1.71i)T + (68.5 - 68.5i)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

−11.39409011757889565563471927983, −10.47776474744047829140186089088, −9.796082439600209822952305908938, −8.607119044540039674795495968373, −8.089590913903604310400722657537, −6.53211028505274154296178906759, −5.65087756316103138372964125429, −4.77552843660092664107534329591, −3.69605154241206494718871792462, −3.24282157514331024793459709928, 1.52242108558737184862374957078, 2.37092671784520788220590253380, 3.40207621309054308589602788246, 4.69000195351767975965409751268, 6.14973529802431092527167900603, 7.01494639686532002284453142717, 7.84188770662085448725282502026, 9.053169549797537822986290360272, 9.936280571477070134046173404169, 11.41137807646777322987400634004

Graph of the ZZ-function along the critical line