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Properties

Label 2-370-185.9-c1-0-12
Degree 22
Conductor 370370
Sign 0.934−0.355i0.934 - 0.355i
Analytic cond. 2.954462.95446
Root an. cond. 1.718851.71885
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  + (−0.984 + 0.173i)2-s + (2.61 + 0.460i)3-s + (0.939 − 0.342i)4-s + (1.38 + 1.75i)5-s − 2.65·6-s + (2.09 − 2.49i)7-s + (−0.866 + 0.5i)8-s + (3.79 + 1.38i)9-s + (−1.66 − 1.48i)10-s + (0.0622 + 0.107i)11-s + (2.61 − 0.460i)12-s + (−2.13 − 5.85i)13-s + (−1.62 + 2.81i)14-s + (2.81 + 5.22i)15-s + (0.766 − 0.642i)16-s + (−1.61 + 4.42i)17-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (1.50 + 0.265i)3-s + (0.469 − 0.171i)4-s + (0.619 + 0.784i)5-s − 1.08·6-s + (0.790 − 0.941i)7-s + (−0.306 + 0.176i)8-s + (1.26 + 0.460i)9-s + (−0.527 − 0.470i)10-s + (0.0187 + 0.0324i)11-s + (0.754 − 0.132i)12-s + (−0.591 − 1.62i)13-s + (−0.434 + 0.753i)14-s + (0.726 + 1.34i)15-s + (0.191 − 0.160i)16-s + (−0.390 + 1.07i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.934−0.355i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.934−0.355i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.934−0.355i0.934 - 0.355i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(9,⋅)\chi_{370} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.934−0.355i)(2,\ 370,\ (\ :1/2),\ 0.934 - 0.355i)

Particular Values

L(1)L(1) ≈\approx 1.75895+0.323265i1.75895 + 0.323265i
L(12)L(\frac12) ≈\approx 1.75895+0.323265i1.75895 + 0.323265i
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+(0.984−0.173i)T 1 + (0.984 - 0.173i)T
5 1+(−1.38−1.75i)T 1 + (-1.38 - 1.75i)T
37 1+(−2.12−5.69i)T 1 + (-2.12 - 5.69i)T
good3 1+(−2.61−0.460i)T+(2.81+1.02i)T2 1 + (-2.61 - 0.460i)T + (2.81 + 1.02i)T^{2}
7 1+(−2.09+2.49i)T+(−1.21−6.89i)T2 1 + (-2.09 + 2.49i)T + (-1.21 - 6.89i)T^{2}
11 1+(−0.0622−0.107i)T+(−5.5+9.52i)T2 1 + (-0.0622 - 0.107i)T + (-5.5 + 9.52i)T^{2}
13 1+(2.13+5.85i)T+(−9.95+8.35i)T2 1 + (2.13 + 5.85i)T + (-9.95 + 8.35i)T^{2}
17 1+(1.61−4.42i)T+(−13.0−10.9i)T2 1 + (1.61 - 4.42i)T + (-13.0 - 10.9i)T^{2}
19 1+(0.307−1.74i)T+(−17.8−6.49i)T2 1 + (0.307 - 1.74i)T + (-17.8 - 6.49i)T^{2}
23 1+(7.34+4.23i)T+(11.5+19.9i)T2 1 + (7.34 + 4.23i)T + (11.5 + 19.9i)T^{2}
29 1+(−1.23−2.13i)T+(−14.5+25.1i)T2 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2}
31 1+5.35T+31T2 1 + 5.35T + 31T^{2}
41 1+(3.81−1.38i)T+(31.4−26.3i)T2 1 + (3.81 - 1.38i)T + (31.4 - 26.3i)T^{2}
43 1+2.22iT−43T2 1 + 2.22iT - 43T^{2}
47 1+(−8.31−4.80i)T+(23.5+40.7i)T2 1 + (-8.31 - 4.80i)T + (23.5 + 40.7i)T^{2}
53 1+(−4.66−5.56i)T+(−9.20+52.1i)T2 1 + (-4.66 - 5.56i)T + (-9.20 + 52.1i)T^{2}
59 1+(−5.60+4.70i)T+(10.2−58.1i)T2 1 + (-5.60 + 4.70i)T + (10.2 - 58.1i)T^{2}
61 1+(−0.127+0.0462i)T+(46.7−39.2i)T2 1 + (-0.127 + 0.0462i)T + (46.7 - 39.2i)T^{2}
67 1+(6.98−8.32i)T+(−11.6−65.9i)T2 1 + (6.98 - 8.32i)T + (-11.6 - 65.9i)T^{2}
71 1+(−1.01+5.74i)T+(−66.7−24.2i)T2 1 + (-1.01 + 5.74i)T + (-66.7 - 24.2i)T^{2}
73 1+9.15iT−73T2 1 + 9.15iT - 73T^{2}
79 1+(−0.898−0.754i)T+(13.7+77.7i)T2 1 + (-0.898 - 0.754i)T + (13.7 + 77.7i)T^{2}
83 1+(1.33−3.65i)T+(−63.5−53.3i)T2 1 + (1.33 - 3.65i)T + (-63.5 - 53.3i)T^{2}
89 1+(11.3−9.50i)T+(15.4−87.6i)T2 1 + (11.3 - 9.50i)T + (15.4 - 87.6i)T^{2}
97 1+(1.40+0.809i)T+(48.5+84.0i)T2 1 + (1.40 + 0.809i)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

−10.81744315302525767448282741697, −10.35952803550849820534173664382, −9.732169199813271068089074807305, −8.465249644412618937050174093366, −7.950642644941947761303482891098, −7.15628505838423070672434262261, −5.81556043884772754359266464452, −4.13449708787836332660187771950, −2.94532079903748788895229529376, −1.84178081612819763346637083878, 1.91640039576920458192777725393, 2.32919952283459082063402924067, 4.16490717381568437128627179879, 5.49602466657445896487993514239, 6.99450601257470302379000467820, 7.918513146723409453580539755628, 8.880875697025969321394912295271, 9.125455717099209767866990462599, 9.897845851207722302704265735314, 11.55669561996159780237266505540

Graph of the ZZ-function along the critical line