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Properties

Label 2-475-19.11-c1-0-19
Degree 22
Conductor 475475
Sign 0.0977+0.995i0.0977 + 0.995i
Analytic cond. 3.792893.79289
Root an. cond. 1.947531.94753
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  + (−1 + 1.73i)2-s + (−0.999 − 1.73i)4-s − 4·7-s + (1.5 + 2.59i)9-s − 11-s + (1 + 1.73i)13-s + (4 − 6.92i)14-s + (1.99 − 3.46i)16-s + (1 − 1.73i)17-s − 6·18-s + (−3.5 − 2.59i)19-s + (1 − 1.73i)22-s + (−3 − 5.19i)23-s − 3.99·26-s + (3.99 + 6.92i)28-s + (−4.5 − 7.79i)29-s + ⋯
L(s)  = 1  + (−0.707 + 1.22i)2-s + (−0.499 − 0.866i)4-s − 1.51·7-s + (0.5 + 0.866i)9-s − 0.301·11-s + (0.277 + 0.480i)13-s + (1.06 − 1.85i)14-s + (0.499 − 0.866i)16-s + (0.242 − 0.420i)17-s − 1.41·18-s + (−0.802 − 0.596i)19-s + (0.213 − 0.369i)22-s + (−0.625 − 1.08i)23-s − 0.784·26-s + (0.755 + 1.30i)28-s + (−0.835 − 1.44i)29-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((0.0977+0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((0.0977+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: 0.0977+0.995i0.0977 + 0.995i
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ475(201,⋅)\chi_{475} (201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 475, ( :1/2), 0.0977+0.995i)(2,\ 475,\ (\ :1/2),\ 0.0977 + 0.995i)

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
19 1+(3.5+2.59i)T 1 + (3.5 + 2.59i)T
good2 1+(1−1.73i)T+(−1−1.73i)T2 1 + (1 - 1.73i)T + (-1 - 1.73i)T^{2}
3 1+(−1.5−2.59i)T2 1 + (-1.5 - 2.59i)T^{2}
7 1+4T+7T2 1 + 4T + 7T^{2}
11 1+T+11T2 1 + T + 11T^{2}
13 1+(−1−1.73i)T+(−6.5+11.2i)T2 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2}
17 1+(−1+1.73i)T+(−8.5−14.7i)T2 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2}
23 1+(3+5.19i)T+(−11.5+19.9i)T2 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2}
29 1+(4.5+7.79i)T+(−14.5+25.1i)T2 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2}
31 1+7T+31T2 1 + 7T + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+(1−1.73i)T+(−20.5−35.5i)T2 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2}
43 1+(1−1.73i)T+(−21.5−37.2i)T2 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2}
47 1+(3+5.19i)T+(−23.5+40.7i)T2 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2}
53 1+(2+3.46i)T+(−26.5+45.8i)T2 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2}
59 1+(4.5−7.79i)T+(−29.5−51.0i)T2 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.5−6.06i)T+(−30.5+52.8i)T2 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5−8.66i)T+(−33.5+58.0i)T2 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2}
71 1+(0.5−0.866i)T+(−35.5−61.4i)T2 1 + (0.5 - 0.866i)T + (-35.5 - 61.4i)T^{2}
73 1+(5−8.66i)T+(−36.5−63.2i)T2 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.5−0.866i)T+(−39.5−68.4i)T2 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2}
83 1+6T+83T2 1 + 6T + 83T^{2}
89 1+(−5.5−9.52i)T+(−44.5+77.0i)T2 1 + (-5.5 - 9.52i)T + (-44.5 + 77.0i)T^{2}
97 1+(−3+5.19i)T+(−48.5−84.0i)T2 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)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

−10.36751164086513074431839345817, −9.716328079682888654030215313233, −8.910473288127111086879216213944, −7.977194035414614945938543856948, −7.07799198823784898065570791569, −6.43332306229834107196970330317, −5.51485640114720601559545485755, −4.10228078569520378231572376488, −2.55355022563390817707266768711, 0, 1.68095685129292630338843942599, 3.29013321991267835158951725753, 3.70399542926455084123582712884, 5.75222278372115589474196927489, 6.56685710101383599199555415969, 7.81680258989507878547397959837, 9.061086989972332851215588596742, 9.532024088411233542618671296969, 10.30600777154511825610365359512

Graph of the ZZ-function along the critical line