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Properties

Label 2-475-19.11-c1-0-23
Degree 22
Conductor 475475
Sign −0.813−0.582i-0.813 - 0.582i
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.14 − 1.97i)2-s + (−1.25 + 2.17i)3-s + (−1.61 − 2.79i)4-s + (2.86 + 4.96i)6-s − 3.50·7-s − 2.79·8-s + (−1.64 − 2.84i)9-s − 4.50·11-s + 8.07·12-s + (−2.5 − 4.33i)13-s + (−4.00 + 6.94i)14-s + (0.0316 − 0.0547i)16-s + (0.0793 − 0.137i)17-s − 7.50·18-s + (−4.26 + 0.920i)19-s + ⋯
L(s)  = 1  + (0.807 − 1.39i)2-s + (−0.723 + 1.25i)3-s + (−0.805 − 1.39i)4-s + (1.16 + 2.02i)6-s − 1.32·7-s − 0.987·8-s + (−0.547 − 0.948i)9-s − 1.35·11-s + 2.33·12-s + (−0.693 − 1.20i)13-s + (−1.07 + 1.85i)14-s + (0.00790 − 0.0136i)16-s + (0.0192 − 0.0333i)17-s − 1.76·18-s + (−0.977 + 0.211i)19-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=((−0.813−0.582i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=((−0.813−0.582i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: −0.813−0.582i-0.813 - 0.582i
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: 00
Selberg data: (2, 475, ( :1/2), −0.813−0.582i)(2,\ 475,\ (\ :1/2),\ -0.813 - 0.582i)

Particular Values

L(1)L(1) ≈\approx 0.0661374+0.205939i0.0661374 + 0.205939i
L(12)L(\frac12) ≈\approx 0.0661374+0.205939i0.0661374 + 0.205939i
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+(4.26−0.920i)T 1 + (4.26 - 0.920i)T
good2 1+(−1.14+1.97i)T+(−1−1.73i)T2 1 + (-1.14 + 1.97i)T + (-1 - 1.73i)T^{2}
3 1+(1.25−2.17i)T+(−1.5−2.59i)T2 1 + (1.25 - 2.17i)T + (-1.5 - 2.59i)T^{2}
7 1+3.50T+7T2 1 + 3.50T + 7T^{2}
11 1+4.50T+11T2 1 + 4.50T + 11T^{2}
13 1+(2.5+4.33i)T+(−6.5+11.2i)T2 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2}
17 1+(−0.0793+0.137i)T+(−8.5−14.7i)T2 1 + (-0.0793 + 0.137i)T + (-8.5 - 14.7i)T^{2}
23 1+(−0.579−1.00i)T+(−11.5+19.9i)T2 1 + (-0.579 - 1.00i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.75−3.03i)T+(−14.5+25.1i)T2 1 + (-1.75 - 3.03i)T + (-14.5 + 25.1i)T^{2}
31 1+2.28T+31T2 1 + 2.28T + 31T^{2}
37 1−10.9T+37T2 1 - 10.9T + 37T^{2}
41 1+(3.03−5.26i)T+(−20.5−35.5i)T2 1 + (3.03 - 5.26i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.67−2.89i)T+(−21.5−37.2i)T2 1 + (1.67 - 2.89i)T + (-21.5 - 37.2i)T^{2}
47 1+(−1.53−2.65i)T+(−23.5+40.7i)T2 1 + (-1.53 - 2.65i)T + (-23.5 + 40.7i)T^{2}
53 1+(2.87+4.97i)T+(−26.5+45.8i)T2 1 + (2.87 + 4.97i)T + (-26.5 + 45.8i)T^{2}
59 1+(1.53−2.65i)T+(−29.5−51.0i)T2 1 + (1.53 - 2.65i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.436−0.756i)T+(−30.5+52.8i)T2 1 + (-0.436 - 0.756i)T + (-30.5 + 52.8i)T^{2}
67 1+(4.22+7.31i)T+(−33.5+58.0i)T2 1 + (4.22 + 7.31i)T + (-33.5 + 58.0i)T^{2}
71 1+(−8.11+14.0i)T+(−35.5−61.4i)T2 1 + (-8.11 + 14.0i)T + (-35.5 - 61.4i)T^{2}
73 1+(3.57−6.19i)T+(−36.5−63.2i)T2 1 + (3.57 - 6.19i)T + (-36.5 - 63.2i)T^{2}
79 1+(−5.06+8.76i)T+(−39.5−68.4i)T2 1 + (-5.06 + 8.76i)T + (-39.5 - 68.4i)T^{2}
83 1+4.85T+83T2 1 + 4.85T + 83T^{2}
89 1+(−0.556−0.963i)T+(−44.5+77.0i)T2 1 + (-0.556 - 0.963i)T + (-44.5 + 77.0i)T^{2}
97 1+(−0.809+1.40i)T+(−48.5−84.0i)T2 1 + (-0.809 + 1.40i)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

−10.55998703799575554561156955048, −10.02042717909538157871654471960, −9.456658743270087684607599663355, −7.82077198152384746434390818703, −6.14872634281463052378911103775, −5.27241373098238115341083553711, −4.56386455995862912981249748125, −3.40688307374145805359966193421, −2.68650354244794469068216419111, −0.10201020570546821500503383326, 2.46877666432954634516220727658, 4.18516220977094387832365994037, 5.35146348392470295361451525187, 6.19030055253175724192564695084, 6.78734650159260270731538044008, 7.38026574699299279418345046448, 8.324767151051877294475340294184, 9.617788465893041322026044962170, 10.81105518472560256017036040510, 12.04268872892607586847846794086

Graph of the ZZ-function along the critical line