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Properties

Label 2-525-21.17-c1-0-42
Degree 22
Conductor 525525
Sign 0.299+0.953i0.299 + 0.953i
Analytic cond. 4.192144.19214
Root an. cond. 2.047472.04747
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.17 − 1.25i)2-s + (1.73 − 0.0401i)3-s + (2.14 − 3.71i)4-s + (3.71 − 2.25i)6-s + (−2.06 + 1.65i)7-s − 5.74i·8-s + (2.99 − 0.138i)9-s + (−1.48 − 0.859i)11-s + (3.56 − 6.51i)12-s + 0.360i·13-s + (−2.39 + 6.18i)14-s + (−2.91 − 5.04i)16-s + (−1.27 + 2.20i)17-s + (6.33 − 4.05i)18-s + (−4.93 + 2.84i)19-s + ⋯
L(s)  = 1  + (1.53 − 0.886i)2-s + (0.999 − 0.0231i)3-s + (1.07 − 1.85i)4-s + (1.51 − 0.922i)6-s + (−0.778 + 0.627i)7-s − 2.03i·8-s + (0.998 − 0.0463i)9-s + (−0.448 − 0.259i)11-s + (1.02 − 1.88i)12-s + 0.100i·13-s + (−0.639 + 1.65i)14-s + (−0.727 − 1.26i)16-s + (−0.308 + 0.534i)17-s + (1.49 − 0.956i)18-s + (−1.13 + 0.653i)19-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((0.299+0.953i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+1/2) L(s)=((0.299+0.953i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: 0.299+0.953i0.299 + 0.953i
Analytic conductor: 4.192144.19214
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ525(101,⋅)\chi_{525} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :1/2), 0.299+0.953i)(2,\ 525,\ (\ :1/2),\ 0.299 + 0.953i)

Particular Values

L(1)L(1) ≈\approx 3.27675−2.40492i3.27675 - 2.40492i
L(12)L(\frac12) ≈\approx 3.27675−2.40492i3.27675 - 2.40492i
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)
bad3 1+(−1.73+0.0401i)T 1 + (-1.73 + 0.0401i)T
5 1 1
7 1+(2.06−1.65i)T 1 + (2.06 - 1.65i)T
good2 1+(−2.17+1.25i)T+(1−1.73i)T2 1 + (-2.17 + 1.25i)T + (1 - 1.73i)T^{2}
11 1+(1.48+0.859i)T+(5.5+9.52i)T2 1 + (1.48 + 0.859i)T + (5.5 + 9.52i)T^{2}
13 1−0.360iT−13T2 1 - 0.360iT - 13T^{2}
17 1+(1.27−2.20i)T+(−8.5−14.7i)T2 1 + (1.27 - 2.20i)T + (-8.5 - 14.7i)T^{2}
19 1+(4.93−2.84i)T+(9.5−16.4i)T2 1 + (4.93 - 2.84i)T + (9.5 - 16.4i)T^{2}
23 1+(−2.17+1.25i)T+(11.5−19.9i)T2 1 + (-2.17 + 1.25i)T + (11.5 - 19.9i)T^{2}
29 1+3.76iT−29T2 1 + 3.76iT - 29T^{2}
31 1+(2.41+1.39i)T+(15.5+26.8i)T2 1 + (2.41 + 1.39i)T + (15.5 + 26.8i)T^{2}
37 1+(−1.65−2.86i)T+(−18.5+32.0i)T2 1 + (-1.65 - 2.86i)T + (-18.5 + 32.0i)T^{2}
41 1−2.63T+41T2 1 - 2.63T + 41T^{2}
43 1+10.0T+43T2 1 + 10.0T + 43T^{2}
47 1+(−2.91−5.04i)T+(−23.5+40.7i)T2 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.25−0.727i)T+(26.5+45.8i)T2 1 + (-1.25 - 0.727i)T + (26.5 + 45.8i)T^{2}
59 1+(3.42−5.93i)T+(−29.5−51.0i)T2 1 + (3.42 - 5.93i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.38+0.801i)T+(30.5−52.8i)T2 1 + (-1.38 + 0.801i)T + (30.5 - 52.8i)T^{2}
67 1+(−1.24+2.15i)T+(−33.5−58.0i)T2 1 + (-1.24 + 2.15i)T + (-33.5 - 58.0i)T^{2}
71 1+13.1iT−71T2 1 + 13.1iT - 71T^{2}
73 1+(−10.0−5.82i)T+(36.5+63.2i)T2 1 + (-10.0 - 5.82i)T + (36.5 + 63.2i)T^{2}
79 1+(6.93+12.0i)T+(−39.5+68.4i)T2 1 + (6.93 + 12.0i)T + (-39.5 + 68.4i)T^{2}
83 1−3.50T+83T2 1 - 3.50T + 83T^{2}
89 1+(6.10+10.5i)T+(−44.5+77.0i)T2 1 + (6.10 + 10.5i)T + (-44.5 + 77.0i)T^{2}
97 1−8.18iT−97T2 1 - 8.18iT - 97T^{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

−10.73980165761843680123791512421, −10.08102621819846121391926308910, −9.074678255009932010821176088701, −8.079006231100045215289956685397, −6.63117014323239046183780987605, −5.89331200067215382736564761518, −4.63204796451176986532227374238, −3.70999783835612700419209549471, −2.82457394652934914484305455289, −1.94057654447273535633215103819, 2.51458170973122781930858160858, 3.46820300929947064145412225328, 4.32609371000911298151044420812, 5.26996317848594373515497977912, 6.69954678419089114741643049721, 7.04490837477303716312941999736, 8.024531091893439168422227140763, 9.047700381337351500507089299426, 10.12901866831140731581980728578, 11.19180758275850718937399546751

Graph of the ZZ-function along the critical line