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Properties

Label 2-370-37.36-c1-0-1
Degree 22
Conductor 370370
Sign 0.525−0.850i0.525 - 0.850i
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  − i·2-s − 1.14·3-s − 4-s + i·5-s + 1.14i·6-s − 0.342·7-s + i·8-s − 1.68·9-s + 10-s + 1.19·11-s + 1.14·12-s + 4.68i·13-s + 0.342i·14-s − 1.14i·15-s + 16-s + 4.17i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.661·3-s − 0.5·4-s + 0.447i·5-s + 0.468i·6-s − 0.129·7-s + 0.353i·8-s − 0.561·9-s + 0.316·10-s + 0.360·11-s + 0.330·12-s + 1.29i·13-s + 0.0916i·14-s − 0.295i·15-s + 0.250·16-s + 1.01i·17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.525−0.850i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.525−0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.525−0.850i0.525 - 0.850i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(221,⋅)\chi_{370} (221, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.525−0.850i)(2,\ 370,\ (\ :1/2),\ 0.525 - 0.850i)

Particular Values

L(1)L(1) ≈\approx 0.617554+0.344413i0.617554 + 0.344413i
L(12)L(\frac12) ≈\approx 0.617554+0.344413i0.617554 + 0.344413i
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+iT 1 + iT
5 1−iT 1 - iT
37 1+(3.19−5.17i)T 1 + (3.19 - 5.17i)T
good3 1+1.14T+3T2 1 + 1.14T + 3T^{2}
7 1+0.342T+7T2 1 + 0.342T + 7T^{2}
11 1−1.19T+11T2 1 - 1.19T + 11T^{2}
13 1−4.68iT−13T2 1 - 4.68iT - 13T^{2}
17 1−4.17iT−17T2 1 - 4.17iT - 17T^{2}
19 1−3.14iT−19T2 1 - 3.14iT - 19T^{2}
23 1−4.68iT−23T2 1 - 4.68iT - 23T^{2}
29 1+0.803iT−29T2 1 + 0.803iT - 29T^{2}
31 1+4.63iT−31T2 1 + 4.63iT - 31T^{2}
41 1−3.78T+41T2 1 - 3.78T + 41T^{2}
43 1−1.19iT−43T2 1 - 1.19iT - 43T^{2}
47 1−3.83T+47T2 1 - 3.83T + 47T^{2}
53 1+11.1T+53T2 1 + 11.1T + 53T^{2}
59 1+0.167iT−59T2 1 + 0.167iT - 59T^{2}
61 1+2.17iT−61T2 1 + 2.17iT - 61T^{2}
67 1+1.24T+67T2 1 + 1.24T + 67T^{2}
71 1−0.685T+71T2 1 - 0.685T + 71T^{2}
73 1+11.9T+73T2 1 + 11.9T + 73T^{2}
79 1+5.14iT−79T2 1 + 5.14iT - 79T^{2}
83 1−4.22T+83T2 1 - 4.22T + 83T^{2}
89 1−4.58iT−89T2 1 - 4.58iT - 89T^{2}
97 1−4.21iT−97T2 1 - 4.21iT - 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

−11.53950805851911405950000659779, −10.84621041198280719911883933654, −9.884372998547044949999666792093, −9.011580884845708462647618050866, −7.899910372832608648497055592126, −6.53999548683079857210553413401, −5.80038710907015613689055800045, −4.44076266777878734100764604922, −3.33995885709405898999953572755, −1.76169980122040947232616136882, 0.52537967101425068786978737934, 3.03382614885322601685958042059, 4.68472604067084461177339499677, 5.43886068796977401790629280949, 6.35135894572237812660437542987, 7.37842996367695302901964751585, 8.467945555306155161104821769906, 9.204454609720456101514015938899, 10.36339021387972400136835650220, 11.24351360592439260654377680822

Graph of the ZZ-function along the critical line