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Properties

Label 2-425-85.19-c1-0-20
Degree 22
Conductor 425425
Sign −0.921−0.387i-0.921 - 0.387i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.292 + 0.292i)2-s + (−1 − 2.41i)3-s + 1.82i·4-s + (1 + 0.414i)6-s + (−1 − 0.414i)7-s + (−1.12 − 1.12i)8-s + (−2.70 + 2.70i)9-s + (−1 − 0.414i)11-s + (4.41 − 1.82i)12-s − 1.41·13-s + (0.414 − 0.171i)14-s − 3·16-s + (−3 + 2.82i)17-s − 1.58i·18-s + (−3.41 − 3.41i)19-s + ⋯
L(s)  = 1  + (−0.207 + 0.207i)2-s + (−0.577 − 1.39i)3-s + 0.914i·4-s + (0.408 + 0.169i)6-s + (−0.377 − 0.156i)7-s + (−0.396 − 0.396i)8-s + (−0.902 + 0.902i)9-s + (−0.301 − 0.124i)11-s + (1.27 − 0.527i)12-s − 0.392·13-s + (0.110 − 0.0458i)14-s − 0.750·16-s + (−0.727 + 0.685i)17-s − 0.373i·18-s + (−0.783 − 0.783i)19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((−0.921−0.387i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((−0.921−0.387i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −0.921−0.387i-0.921 - 0.387i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(274,⋅)\chi_{425} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 425, ( :1/2), −0.921−0.387i)(2,\ 425,\ (\ :1/2),\ -0.921 - 0.387i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+(3−2.82i)T 1 + (3 - 2.82i)T
good2 1+(0.292−0.292i)T−2iT2 1 + (0.292 - 0.292i)T - 2iT^{2}
3 1+(1+2.41i)T+(−2.12+2.12i)T2 1 + (1 + 2.41i)T + (-2.12 + 2.12i)T^{2}
7 1+(1+0.414i)T+(4.94+4.94i)T2 1 + (1 + 0.414i)T + (4.94 + 4.94i)T^{2}
11 1+(1+0.414i)T+(7.77+7.77i)T2 1 + (1 + 0.414i)T + (7.77 + 7.77i)T^{2}
13 1+1.41T+13T2 1 + 1.41T + 13T^{2}
19 1+(3.41+3.41i)T+19iT2 1 + (3.41 + 3.41i)T + 19iT^{2}
23 1+(1.58−3.82i)T+(−16.2−16.2i)T2 1 + (1.58 - 3.82i)T + (-16.2 - 16.2i)T^{2}
29 1+(−1.70−4.12i)T+(−20.5+20.5i)T2 1 + (-1.70 - 4.12i)T + (-20.5 + 20.5i)T^{2}
31 1+(3−1.24i)T+(21.9−21.9i)T2 1 + (3 - 1.24i)T + (21.9 - 21.9i)T^{2}
37 1+(−1.46−3.53i)T+(−26.1+26.1i)T2 1 + (-1.46 - 3.53i)T + (-26.1 + 26.1i)T^{2}
41 1+(3.12−7.53i)T+(−28.9−28.9i)T2 1 + (3.12 - 7.53i)T + (-28.9 - 28.9i)T^{2}
43 1+(3.41+3.41i)T+43iT2 1 + (3.41 + 3.41i)T + 43iT^{2}
47 1+10.8T+47T2 1 + 10.8T + 47T^{2}
53 1+(−1+i)T−53iT2 1 + (-1 + i)T - 53iT^{2}
59 1+(−4.24+4.24i)T−59iT2 1 + (-4.24 + 4.24i)T - 59iT^{2}
61 1+(−3.53+8.53i)T+(−43.1−43.1i)T2 1 + (-3.53 + 8.53i)T + (-43.1 - 43.1i)T^{2}
67 1+6.82iT−67T2 1 + 6.82iT - 67T^{2}
71 1+(−12.0+5i)T+(50.2−50.2i)T2 1 + (-12.0 + 5i)T + (50.2 - 50.2i)T^{2}
73 1+(−4.94+2.05i)T+(51.6−51.6i)T2 1 + (-4.94 + 2.05i)T + (51.6 - 51.6i)T^{2}
79 1+(−3.82−1.58i)T+(55.8+55.8i)T2 1 + (-3.82 - 1.58i)T + (55.8 + 55.8i)T^{2}
83 1+(−0.242+0.242i)T−83iT2 1 + (-0.242 + 0.242i)T - 83iT^{2}
89 1+9.41iT−89T2 1 + 9.41iT - 89T^{2}
97 1+(−5.94+2.46i)T+(68.5−68.5i)T2 1 + (-5.94 + 2.46i)T + (68.5 - 68.5i)T^{2}
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   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

−11.00275154066812877708083506156, −9.650605503500711633510530476798, −8.461712094088775646650720759822, −7.84276778428535037952558298962, −6.74139466625076497267501010283, −6.51525938650821944907948057307, −4.96999559640665771319244717240, −3.42004887698769473364810613663, −2.01830063658283705206385857218, 0, 2.39289955090906246927740323385, 4.05241045658196203350523733790, 4.95124961730577736204645377171, 5.78602528569735038141163518109, 6.72645989616945193200399518182, 8.395170124393985841844533636386, 9.438369382461915981316549295987, 9.927280108828869419373807776074, 10.65062340727806412462916462046

Graph of the ZZ-function along the critical line