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Properties

Label 2-370-37.36-c1-0-4
Degree 22
Conductor 370370
Sign −0.479+0.877i-0.479 + 0.877i
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 − 3.10·3-s − 4-s + i·5-s + 3.10i·6-s + 3.81·7-s + i·8-s + 6.62·9-s + 10-s − 4.91·11-s + 3.10·12-s − 3.62i·13-s − 3.81i·14-s − 3.10i·15-s + 16-s − 6.33i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.79·3-s − 0.5·4-s + 0.447i·5-s + 1.26i·6-s + 1.44·7-s + 0.353i·8-s + 2.20·9-s + 0.316·10-s − 1.48·11-s + 0.895·12-s − 1.00i·13-s − 1.01i·14-s − 0.801i·15-s + 0.250·16-s − 1.53i·17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((−0.479+0.877i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((−0.479+0.877i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: −0.479+0.877i-0.479 + 0.877i
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), −0.479+0.877i)(2,\ 370,\ (\ :1/2),\ -0.479 + 0.877i)

Particular Values

L(1)L(1) ≈\approx 0.316475−0.533530i0.316475 - 0.533530i
L(12)L(\frac12) ≈\approx 0.316475−0.533530i0.316475 - 0.533530i
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+(−2.91+5.33i)T 1 + (-2.91 + 5.33i)T
good3 1+3.10T+3T2 1 + 3.10T + 3T^{2}
7 1−3.81T+7T2 1 - 3.81T + 7T^{2}
11 1+4.91T+11T2 1 + 4.91T + 11T^{2}
13 1+3.62iT−13T2 1 + 3.62iT - 13T^{2}
17 1+6.33iT−17T2 1 + 6.33iT - 17T^{2}
19 1−5.10iT−19T2 1 - 5.10iT - 19T^{2}
23 1+3.62iT−23T2 1 + 3.62iT - 23T^{2}
29 1+6.91iT−29T2 1 + 6.91iT - 29T^{2}
31 1+4.39iT−31T2 1 + 4.39iT - 31T^{2}
41 1−5.49T+41T2 1 - 5.49T + 41T^{2}
43 1+4.91iT−43T2 1 + 4.91iT - 43T^{2}
47 1+2.52T+47T2 1 + 2.52T + 47T^{2}
53 1−3.75T+53T2 1 - 3.75T + 53T^{2}
59 1+6.52iT−59T2 1 + 6.52iT - 59T^{2}
61 1−8.33iT−61T2 1 - 8.33iT - 61T^{2}
67 1−12.9T+67T2 1 - 12.9T + 67T^{2}
71 1+7.62T+71T2 1 + 7.62T + 71T^{2}
73 1+3.15T+73T2 1 + 3.15T + 73T^{2}
79 1+7.10iT−79T2 1 + 7.10iT - 79T^{2}
83 1+14.3T+83T2 1 + 14.3T + 83T^{2}
89 1−12.4iT−89T2 1 - 12.4iT - 89T^{2}
97 1−2.50iT−97T2 1 - 2.50iT - 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

−11.14461613054728385379678841371, −10.51013538419584568348684618825, −9.885020748919131335432174170338, −8.058767809257064262936362077618, −7.41880249886794775960413903521, −5.80410064147084335077337122192, −5.25489651537270955427242124003, −4.34378041327204859113369277379, −2.38149182456606463498627883928, −0.57253874518889279046480561353, 1.44721676661157891359357735599, 4.44689520430763720914960289870, 5.00624569012031963966616369741, 5.72554692809842502880262909603, 6.83604696966976962959528422791, 7.77088713300587005319312080459, 8.730665851077557588114303916424, 10.11952955718796191752854520179, 10.99334284549834502875723172831, 11.48131579491141587861620989376

Graph of the ZZ-function along the critical line