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Properties

Label 2-370-37.16-c1-0-8
Degree 22
Conductor 370370
Sign −0.0216+0.999i-0.0216 + 0.999i
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  + (−0.939 − 0.342i)2-s + (−0.436 + 0.159i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + 0.465·6-s + (0.593 − 3.36i)7-s + (−0.500 − 0.866i)8-s + (−2.13 + 1.78i)9-s + (0.5 − 0.866i)10-s + (−3.22 − 5.58i)11-s + (−0.436 − 0.159i)12-s + (3.08 + 2.58i)13-s + (−1.70 + 2.95i)14-s + (−0.0807 − 0.457i)15-s + (0.173 + 0.984i)16-s + (2.55 − 2.14i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.252 + 0.0918i)3-s + (0.383 + 0.321i)4-s + (−0.0776 + 0.440i)5-s + 0.189·6-s + (0.224 − 1.27i)7-s + (−0.176 − 0.306i)8-s + (−0.710 + 0.596i)9-s + (0.158 − 0.273i)10-s + (−0.972 − 1.68i)11-s + (−0.126 − 0.0459i)12-s + (0.855 + 0.717i)13-s + (−0.456 + 0.790i)14-s + (−0.0208 − 0.118i)15-s + (0.0434 + 0.246i)16-s + (0.620 − 0.520i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((−0.0216+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0216 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((−0.0216+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0216 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: −0.0216+0.999i-0.0216 + 0.999i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(201,⋅)\chi_{370} (201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), −0.0216+0.999i)(2,\ 370,\ (\ :1/2),\ -0.0216 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.502459−0.513456i0.502459 - 0.513456i
L(12)L(\frac12) ≈\approx 0.502459−0.513456i0.502459 - 0.513456i
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+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
5 1+(0.173−0.984i)T 1 + (0.173 - 0.984i)T
37 1+(3.52+4.95i)T 1 + (3.52 + 4.95i)T
good3 1+(0.436−0.159i)T+(2.29−1.92i)T2 1 + (0.436 - 0.159i)T + (2.29 - 1.92i)T^{2}
7 1+(−0.593+3.36i)T+(−6.57−2.39i)T2 1 + (-0.593 + 3.36i)T + (-6.57 - 2.39i)T^{2}
11 1+(3.22+5.58i)T+(−5.5+9.52i)T2 1 + (3.22 + 5.58i)T + (-5.5 + 9.52i)T^{2}
13 1+(−3.08−2.58i)T+(2.25+12.8i)T2 1 + (-3.08 - 2.58i)T + (2.25 + 12.8i)T^{2}
17 1+(−2.55+2.14i)T+(2.95−16.7i)T2 1 + (-2.55 + 2.14i)T + (2.95 - 16.7i)T^{2}
19 1+(−1.82+0.662i)T+(14.5−12.2i)T2 1 + (-1.82 + 0.662i)T + (14.5 - 12.2i)T^{2}
23 1+(−3.97+6.88i)T+(−11.5−19.9i)T2 1 + (-3.97 + 6.88i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.45+7.71i)T+(−14.5+25.1i)T2 1 + (4.45 + 7.71i)T + (-14.5 + 25.1i)T^{2}
31 1+4.45T+31T2 1 + 4.45T + 31T^{2}
41 1+(−2.06−1.73i)T+(7.11+40.3i)T2 1 + (-2.06 - 1.73i)T + (7.11 + 40.3i)T^{2}
43 1−5.37T+43T2 1 - 5.37T + 43T^{2}
47 1+(4.77−8.26i)T+(−23.5−40.7i)T2 1 + (4.77 - 8.26i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.523+2.96i)T+(−49.8+18.1i)T2 1 + (0.523 + 2.96i)T + (-49.8 + 18.1i)T^{2}
59 1+(0.145+0.824i)T+(−55.4+20.1i)T2 1 + (0.145 + 0.824i)T + (-55.4 + 20.1i)T^{2}
61 1+(−2.44−2.04i)T+(10.5+60.0i)T2 1 + (-2.44 - 2.04i)T + (10.5 + 60.0i)T^{2}
67 1+(−0.828+4.69i)T+(−62.9−22.9i)T2 1 + (-0.828 + 4.69i)T + (-62.9 - 22.9i)T^{2}
71 1+(−5.83+2.12i)T+(54.3−45.6i)T2 1 + (-5.83 + 2.12i)T + (54.3 - 45.6i)T^{2}
73 1−11.3T+73T2 1 - 11.3T + 73T^{2}
79 1+(2.17−12.3i)T+(−74.2−27.0i)T2 1 + (2.17 - 12.3i)T + (-74.2 - 27.0i)T^{2}
83 1+(5.12−4.29i)T+(14.4−81.7i)T2 1 + (5.12 - 4.29i)T + (14.4 - 81.7i)T^{2}
89 1+(0.0213+0.121i)T+(−83.6+30.4i)T2 1 + (0.0213 + 0.121i)T + (-83.6 + 30.4i)T^{2}
97 1+(3.31−5.74i)T+(−48.5−84.0i)T2 1 + (3.31 - 5.74i)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

−10.87815671284538619134491355795, −10.72699767071200686267306380273, −9.366696161968068302295114416645, −8.255110136478101886656921705140, −7.66097932309995832100064516800, −6.48813069123643767229763002685, −5.39976560098598996307728070234, −3.85762411464769587660311433143, −2.70700301349335643888111249516, −0.63301431574460952776559645198, 1.68004025274204825913817268634, 3.23885727794462141764908250577, 5.36787892550354934939439470301, 5.53698044289636938064350766188, 7.06736238182779155764422799073, 8.014129681662082458346822204060, 8.881702931830910520830450284531, 9.582823777831799778896829743296, 10.67085117084674213856664509511, 11.62426465879838572806151469499

Graph of the ZZ-function along the critical line