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Properties

Label 2-370-5.4-c1-0-12
Degree 22
Conductor 370370
Sign 0.762+0.646i0.762 + 0.646i
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 − 2.62i·3-s − 4-s + (1.70 + 1.44i)5-s + 2.62·6-s − 1.83i·7-s − i·8-s − 3.89·9-s + (−1.44 + 1.70i)10-s + 4.19·11-s + 2.62i·12-s + 0.369i·13-s + 1.83·14-s + (3.79 − 4.47i)15-s + 16-s − 5.08i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.51i·3-s − 0.5·4-s + (0.762 + 0.646i)5-s + 1.07·6-s − 0.692i·7-s − 0.353i·8-s − 1.29·9-s + (−0.457 + 0.539i)10-s + 1.26·11-s + 0.757i·12-s + 0.102i·13-s + 0.489·14-s + (0.980 − 1.15i)15-s + 0.250·16-s − 1.23i·17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.762+0.646i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.762+0.646i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.762+0.646i0.762 + 0.646i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(149,⋅)\chi_{370} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.762+0.646i)(2,\ 370,\ (\ :1/2),\ 0.762 + 0.646i)

Particular Values

L(1)L(1) ≈\approx 1.37764−0.505616i1.37764 - 0.505616i
L(12)L(\frac12) ≈\approx 1.37764−0.505616i1.37764 - 0.505616i
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+(−1.70−1.44i)T 1 + (-1.70 - 1.44i)T
37 1−iT 1 - iT
good3 1+2.62iT−3T2 1 + 2.62iT - 3T^{2}
7 1+1.83iT−7T2 1 + 1.83iT - 7T^{2}
11 1−4.19T+11T2 1 - 4.19T + 11T^{2}
13 1−0.369iT−13T2 1 - 0.369iT - 13T^{2}
17 1+5.08iT−17T2 1 + 5.08iT - 17T^{2}
19 1+3.55T+19T2 1 + 3.55T + 19T^{2}
23 1+5.62iT−23T2 1 + 5.62iT - 23T^{2}
29 1+1.20T+29T2 1 + 1.20T + 29T^{2}
31 1−10.1T+31T2 1 - 10.1T + 31T^{2}
41 1+8.01T+41T2 1 + 8.01T + 41T^{2}
43 1−2.27iT−43T2 1 - 2.27iT - 43T^{2}
47 1−10.9iT−47T2 1 - 10.9iT - 47T^{2}
53 1−9.94iT−53T2 1 - 9.94iT - 53T^{2}
59 1−5.34T+59T2 1 - 5.34T + 59T^{2}
61 1+9.79T+61T2 1 + 9.79T + 61T^{2}
67 1−1.85iT−67T2 1 - 1.85iT - 67T^{2}
71 1−2.86T+71T2 1 - 2.86T + 71T^{2}
73 1−8.09iT−73T2 1 - 8.09iT - 73T^{2}
79 1+6.06T+79T2 1 + 6.06T + 79T^{2}
83 1−8.93iT−83T2 1 - 8.93iT - 83T^{2}
89 1+11.4T+89T2 1 + 11.4T + 89T^{2}
97 1−6.05iT−97T2 1 - 6.05iT - 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

−11.47529982634392767198273018490, −10.30885637107653620888659353485, −9.269728557592769380594043836003, −8.238062434702185586076647911974, −7.15177947766005251355371343765, −6.67693257951949566975457504481, −6.06058979012472761341266422193, −4.45030878689047533762249394042, −2.68946954117892575433683834986, −1.15742013890095378258044901024, 1.84213289369450174461901842888, 3.49608328756568861810200040506, 4.39486599754975174503097917282, 5.36560315906133467785132791607, 6.29396687202998277697633103484, 8.550835254716635621981619987209, 8.892122956925646261534946046606, 9.878054097338939385932952051973, 10.27921196098941964021646622906, 11.45242072300559178676905444079

Graph of the ZZ-function along the critical line