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Properties

Label 2-476-119.100-c1-0-0
Degree 22
Conductor 476476
Sign −0.998+0.0572i-0.998 + 0.0572i
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  + (−0.940 + 1.22i)3-s + (0.338 − 2.57i)5-s + (−1.75 − 1.97i)7-s + (0.158 + 0.592i)9-s + (−4.68 + 0.616i)11-s + 3.87i·13-s + (2.83 + 2.83i)15-s + (−3.27 + 2.49i)17-s + (−6.62 + 1.77i)19-s + (4.07 − 0.298i)21-s + (−1.66 − 2.16i)23-s + (−1.68 − 0.451i)25-s + (−5.15 − 2.13i)27-s + (0.556 − 0.230i)29-s + (4.31 − 5.62i)31-s + ⋯
L(s)  = 1  + (−0.543 + 0.707i)3-s + (0.151 − 1.15i)5-s + (−0.665 − 0.746i)7-s + (0.0528 + 0.197i)9-s + (−1.41 + 0.186i)11-s + 1.07i·13-s + (0.732 + 0.732i)15-s + (−0.795 + 0.606i)17-s + (−1.51 + 0.407i)19-s + (0.889 − 0.0652i)21-s + (−0.346 − 0.451i)23-s + (−0.336 − 0.0902i)25-s + (−0.992 − 0.411i)27-s + (0.103 − 0.0428i)29-s + (0.775 − 1.01i)31-s + ⋯

Functional equation

Λ(s)=(476s/2 ΓC(s) L(s)=((−0.998+0.0572i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0572i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(476s/2 ΓC(s+1/2) L(s)=((−0.998+0.0572i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0572i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 476476    =    22⋅7⋅172^{2} \cdot 7 \cdot 17
Sign: −0.998+0.0572i-0.998 + 0.0572i
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), −0.998+0.0572i)(2,\ 476,\ (\ :1/2),\ -0.998 + 0.0572i)

Particular Values

L(1)L(1) ≈\approx 0.000801771−0.0279652i0.000801771 - 0.0279652i
L(12)L(\frac12) ≈\approx 0.000801771−0.0279652i0.000801771 - 0.0279652i
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+(1.75+1.97i)T 1 + (1.75 + 1.97i)T
17 1+(3.27−2.49i)T 1 + (3.27 - 2.49i)T
good3 1+(0.940−1.22i)T+(−0.776−2.89i)T2 1 + (0.940 - 1.22i)T + (-0.776 - 2.89i)T^{2}
5 1+(−0.338+2.57i)T+(−4.82−1.29i)T2 1 + (-0.338 + 2.57i)T + (-4.82 - 1.29i)T^{2}
11 1+(4.68−0.616i)T+(10.6−2.84i)T2 1 + (4.68 - 0.616i)T + (10.6 - 2.84i)T^{2}
13 1−3.87iT−13T2 1 - 3.87iT - 13T^{2}
19 1+(6.62−1.77i)T+(16.4−9.5i)T2 1 + (6.62 - 1.77i)T + (16.4 - 9.5i)T^{2}
23 1+(1.66+2.16i)T+(−5.95+22.2i)T2 1 + (1.66 + 2.16i)T + (-5.95 + 22.2i)T^{2}
29 1+(−0.556+0.230i)T+(20.5−20.5i)T2 1 + (-0.556 + 0.230i)T + (20.5 - 20.5i)T^{2}
31 1+(−4.31+5.62i)T+(−8.02−29.9i)T2 1 + (-4.31 + 5.62i)T + (-8.02 - 29.9i)T^{2}
37 1+(0.582+0.0767i)T+(35.7+9.57i)T2 1 + (0.582 + 0.0767i)T + (35.7 + 9.57i)T^{2}
41 1+(2.37+0.983i)T+(28.9+28.9i)T2 1 + (2.37 + 0.983i)T + (28.9 + 28.9i)T^{2}
43 1+(−2.76+2.76i)T−43iT2 1 + (-2.76 + 2.76i)T - 43iT^{2}
47 1+(6.58+3.80i)T+(23.5+40.7i)T2 1 + (6.58 + 3.80i)T + (23.5 + 40.7i)T^{2}
53 1+(−0.299+1.11i)T+(−45.8−26.5i)T2 1 + (-0.299 + 1.11i)T + (-45.8 - 26.5i)T^{2}
59 1+(−4.18−1.12i)T+(51.0+29.5i)T2 1 + (-4.18 - 1.12i)T + (51.0 + 29.5i)T^{2}
61 1+(4.93−3.78i)T+(15.7−58.9i)T2 1 + (4.93 - 3.78i)T + (15.7 - 58.9i)T^{2}
67 1+(−4.69−8.13i)T+(−33.5+58.0i)T2 1 + (-4.69 - 8.13i)T + (-33.5 + 58.0i)T^{2}
71 1+(−2.13−5.14i)T+(−50.2+50.2i)T2 1 + (-2.13 - 5.14i)T + (-50.2 + 50.2i)T^{2}
73 1+(11.5+8.83i)T+(18.8+70.5i)T2 1 + (11.5 + 8.83i)T + (18.8 + 70.5i)T^{2}
79 1+(−8.98−11.7i)T+(−20.4+76.3i)T2 1 + (-8.98 - 11.7i)T + (-20.4 + 76.3i)T^{2}
83 1+(7.85+7.85i)T+83iT2 1 + (7.85 + 7.85i)T + 83iT^{2}
89 1+(−1.39−0.804i)T+(44.5+77.0i)T2 1 + (-1.39 - 0.804i)T + (44.5 + 77.0i)T^{2}
97 1+(−5.18+2.14i)T+(68.5−68.5i)T2 1 + (-5.18 + 2.14i)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

−11.25065108136688450472399747253, −10.41073718582092815729432871926, −9.941147491257308083901512721380, −8.816337740057683444836683768817, −8.014390225439293269341237548624, −6.71765677099911296742880503127, −5.69868277852780786716606226635, −4.56773476126131881722921759012, −4.17796526721909571668914145052, −2.13131841374912956444530893869, 0.01662003830731141878320007024, 2.40885191883332549884006049288, 3.18218620293421265391887346180, 5.06775240396509419227842600618, 6.16378583726273462949773444209, 6.61950207032494742151640092469, 7.62935175726852238244018786011, 8.661753743756063447931267935831, 9.893283119074538438194347482456, 10.64852323806582967811649986176

Graph of the ZZ-function along the critical line