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Properties

Label 2-370-185.184-c1-0-8
Degree 22
Conductor 370370
Sign 0.968−0.250i0.968 - 0.250i
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  + 2-s − 0.377i·3-s + 4-s + (−1.04 + 1.97i)5-s − 0.377i·6-s − 0.631i·7-s + 8-s + 2.85·9-s + (−1.04 + 1.97i)10-s + 1.24·11-s − 0.377i·12-s + 3.34·13-s − 0.631i·14-s + (0.746 + 0.395i)15-s + 16-s + 3.10·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.218i·3-s + 0.5·4-s + (−0.468 + 0.883i)5-s − 0.154i·6-s − 0.238i·7-s + 0.353·8-s + 0.952·9-s + (−0.331 + 0.624i)10-s + 0.376·11-s − 0.109i·12-s + 0.929·13-s − 0.168i·14-s + (0.192 + 0.102i)15-s + 0.250·16-s + 0.753·17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.968−0.250i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.968−0.250i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.968−0.250i0.968 - 0.250i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(369,⋅)\chi_{370} (369, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.968−0.250i)(2,\ 370,\ (\ :1/2),\ 0.968 - 0.250i)

Particular Values

L(1)L(1) ≈\approx 2.04902+0.260611i2.04902 + 0.260611i
L(12)L(\frac12) ≈\approx 2.04902+0.260611i2.04902 + 0.260611i
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−T 1 - T
5 1+(1.04−1.97i)T 1 + (1.04 - 1.97i)T
37 1+(4.10+4.48i)T 1 + (4.10 + 4.48i)T
good3 1+0.377iT−3T2 1 + 0.377iT - 3T^{2}
7 1+0.631iT−7T2 1 + 0.631iT - 7T^{2}
11 1−1.24T+11T2 1 - 1.24T + 11T^{2}
13 1−3.34T+13T2 1 - 3.34T + 13T^{2}
17 1−3.10T+17T2 1 - 3.10T + 17T^{2}
19 1−5.97iT−19T2 1 - 5.97iT - 19T^{2}
23 1+7.60T+23T2 1 + 7.60T + 23T^{2}
29 1+9.57iT−29T2 1 + 9.57iT - 29T^{2}
31 1−7.26iT−31T2 1 - 7.26iT - 31T^{2}
41 1+8.45T+41T2 1 + 8.45T + 41T^{2}
43 1+4.86T+43T2 1 + 4.86T + 43T^{2}
47 1+13.1iT−47T2 1 + 13.1iT - 47T^{2}
53 1−7.17iT−53T2 1 - 7.17iT - 53T^{2}
59 1+4.36iT−59T2 1 + 4.36iT - 59T^{2}
61 1+2.14iT−61T2 1 + 2.14iT - 61T^{2}
67 1−11.3iT−67T2 1 - 11.3iT - 67T^{2}
71 1+12.7T+71T2 1 + 12.7T + 71T^{2}
73 1−4.45iT−73T2 1 - 4.45iT - 73T^{2}
79 1+8.78iT−79T2 1 + 8.78iT - 79T^{2}
83 1−6.63iT−83T2 1 - 6.63iT - 83T^{2}
89 1+13.7iT−89T2 1 + 13.7iT - 89T^{2}
97 1−11.0T+97T2 1 - 11.0T + 97T^{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

−11.86391605977728200880053630706, −10.33928670801098963741726916937, −10.17550667955299126702811610654, −8.331387648450424906346474109041, −7.50135563478597676652581842105, −6.60959792589456473226612287299, −5.76357120815864462371683480803, −4.08532625308258404663766860404, −3.55946677321607037982958348439, −1.79158704786999929262269630690, 1.49268974262983817099086377144, 3.45715606246204618043320582855, 4.37067580394668454147739842349, 5.27384187098563786199027792902, 6.45270859264557915541434531324, 7.55072336993694859881495665213, 8.573846417888219642703611588560, 9.514461157426103623943952256949, 10.57772688564403866369237988595, 11.62629197279110086425948792138

Graph of the ZZ-function along the critical line