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Properties

Label 2-476-119.100-c1-0-10
Degree 22
Conductor 476476
Sign −0.0362+0.999i-0.0362 + 0.999i
Analytic cond. 3.800873.80087
Root an. cond. 1.949581.94958
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  + (1.75 − 2.28i)3-s + (−0.0892 + 0.678i)5-s + (1.39 − 2.25i)7-s + (−1.37 − 5.12i)9-s + (−2.70 + 0.355i)11-s − 1.91i·13-s + (1.39 + 1.39i)15-s + (3.97 − 1.09i)17-s + (−7.44 + 1.99i)19-s + (−2.70 − 7.12i)21-s + (3.96 + 5.16i)23-s + (4.37 + 1.17i)25-s + (−6.14 − 2.54i)27-s + (8.44 − 3.49i)29-s + (0.277 − 0.361i)31-s + ⋯
L(s)  = 1  + (1.01 − 1.32i)3-s + (−0.0399 + 0.303i)5-s + (0.525 − 0.850i)7-s + (−0.457 − 1.70i)9-s + (−0.814 + 0.107i)11-s − 0.530i·13-s + (0.359 + 0.359i)15-s + (0.964 − 0.264i)17-s + (−1.70 + 0.457i)19-s + (−0.590 − 1.55i)21-s + (0.826 + 1.07i)23-s + (0.875 + 0.234i)25-s + (−1.18 − 0.489i)27-s + (1.56 − 0.649i)29-s + (0.0497 − 0.0648i)31-s + ⋯

Functional equation

Λ(s)=(476s/2 ΓC(s) L(s)=((−0.0362+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0362 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(476s/2 ΓC(s+1/2) L(s)=((−0.0362+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0362 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 476476    =    22⋅7⋅172^{2} \cdot 7 \cdot 17
Sign: −0.0362+0.999i-0.0362 + 0.999i
Analytic conductor: 3.800873.80087
Root analytic conductor: 1.949581.94958
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ476(457,⋅)\chi_{476} (457, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 476, ( :1/2), −0.0362+0.999i)(2,\ 476,\ (\ :1/2),\ -0.0362 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.33768−1.38706i1.33768 - 1.38706i
L(12)L(\frac12) ≈\approx 1.33768−1.38706i1.33768 - 1.38706i
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)
bad2 1 1
7 1+(−1.39+2.25i)T 1 + (-1.39 + 2.25i)T
17 1+(−3.97+1.09i)T 1 + (-3.97 + 1.09i)T
good3 1+(−1.75+2.28i)T+(−0.776−2.89i)T2 1 + (-1.75 + 2.28i)T + (-0.776 - 2.89i)T^{2}
5 1+(0.0892−0.678i)T+(−4.82−1.29i)T2 1 + (0.0892 - 0.678i)T + (-4.82 - 1.29i)T^{2}
11 1+(2.70−0.355i)T+(10.6−2.84i)T2 1 + (2.70 - 0.355i)T + (10.6 - 2.84i)T^{2}
13 1+1.91iT−13T2 1 + 1.91iT - 13T^{2}
19 1+(7.44−1.99i)T+(16.4−9.5i)T2 1 + (7.44 - 1.99i)T + (16.4 - 9.5i)T^{2}
23 1+(−3.96−5.16i)T+(−5.95+22.2i)T2 1 + (-3.96 - 5.16i)T + (-5.95 + 22.2i)T^{2}
29 1+(−8.44+3.49i)T+(20.5−20.5i)T2 1 + (-8.44 + 3.49i)T + (20.5 - 20.5i)T^{2}
31 1+(−0.277+0.361i)T+(−8.02−29.9i)T2 1 + (-0.277 + 0.361i)T + (-8.02 - 29.9i)T^{2}
37 1+(7.40+0.974i)T+(35.7+9.57i)T2 1 + (7.40 + 0.974i)T + (35.7 + 9.57i)T^{2}
41 1+(−3.96−1.64i)T+(28.9+28.9i)T2 1 + (-3.96 - 1.64i)T + (28.9 + 28.9i)T^{2}
43 1+(5.63−5.63i)T−43iT2 1 + (5.63 - 5.63i)T - 43iT^{2}
47 1+(7.20+4.15i)T+(23.5+40.7i)T2 1 + (7.20 + 4.15i)T + (23.5 + 40.7i)T^{2}
53 1+(0.206−0.769i)T+(−45.8−26.5i)T2 1 + (0.206 - 0.769i)T + (-45.8 - 26.5i)T^{2}
59 1+(−5.80−1.55i)T+(51.0+29.5i)T2 1 + (-5.80 - 1.55i)T + (51.0 + 29.5i)T^{2}
61 1+(0.639−0.490i)T+(15.7−58.9i)T2 1 + (0.639 - 0.490i)T + (15.7 - 58.9i)T^{2}
67 1+(−7.59−13.1i)T+(−33.5+58.0i)T2 1 + (-7.59 - 13.1i)T + (-33.5 + 58.0i)T^{2}
71 1+(−0.336−0.811i)T+(−50.2+50.2i)T2 1 + (-0.336 - 0.811i)T + (-50.2 + 50.2i)T^{2}
73 1+(−2.84−2.18i)T+(18.8+70.5i)T2 1 + (-2.84 - 2.18i)T + (18.8 + 70.5i)T^{2}
79 1+(−2.21−2.88i)T+(−20.4+76.3i)T2 1 + (-2.21 - 2.88i)T + (-20.4 + 76.3i)T^{2}
83 1+(−3.81−3.81i)T+83iT2 1 + (-3.81 - 3.81i)T + 83iT^{2}
89 1+(−9.12−5.26i)T+(44.5+77.0i)T2 1 + (-9.12 - 5.26i)T + (44.5 + 77.0i)T^{2}
97 1+(−5.24+2.17i)T+(68.5−68.5i)T2 1 + (-5.24 + 2.17i)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

−10.71018981877957179046010833340, −9.954093823157116398638066756471, −8.516361613301924509827136461629, −8.039278842208090957160829417899, −7.26889529840309537712966186976, −6.51702470781619755607528185970, −5.06394139542098445800976217164, −3.53123077851181125683282498546, −2.51108447985337794125602846700, −1.17017680419381091252401643237, 2.28215674561093583323396586153, 3.25989524362416185735288446560, 4.65554135615827542975984710606, 5.04678280743605357011367784431, 6.58636925678063558965064983740, 8.251319626985917913175580854956, 8.521871735493402208642980654536, 9.266978257663712932945933365799, 10.44763950557269514811651282234, 10.76416261321629581108232290328

Graph of the ZZ-function along the critical line