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Properties

Label 2-370-185.68-c1-0-6
Degree 22
Conductor 370370
Sign 0.309−0.950i0.309 - 0.950i
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 + 4-s + (−1 + 2i)5-s + (−2 + 2i)7-s + 8-s + 3i·9-s + (−1 + 2i)10-s + 4·13-s + (−2 + 2i)14-s + 16-s + 2i·17-s + 3i·18-s + (−2 − 2i)19-s + (−1 + 2i)20-s + 4·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.447 + 0.894i)5-s + (−0.755 + 0.755i)7-s + 0.353·8-s + i·9-s + (−0.316 + 0.632i)10-s + 1.10·13-s + (−0.534 + 0.534i)14-s + 0.250·16-s + 0.485i·17-s + 0.707i·18-s + (−0.458 − 0.458i)19-s + (−0.223 + 0.447i)20-s + 0.834·23-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.309−0.950i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.309−0.950i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.309−0.950i0.309 - 0.950i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(253,⋅)\chi_{370} (253, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.309−0.950i)(2,\ 370,\ (\ :1/2),\ 0.309 - 0.950i)

Particular Values

L(1)L(1) ≈\approx 1.42359+1.03404i1.42359 + 1.03404i
L(12)L(\frac12) ≈\approx 1.42359+1.03404i1.42359 + 1.03404i
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+(1−2i)T 1 + (1 - 2i)T
37 1+(1−6i)T 1 + (1 - 6i)T
good3 1−3iT2 1 - 3iT^{2}
7 1+(2−2i)T−7iT2 1 + (2 - 2i)T - 7iT^{2}
11 1−11T2 1 - 11T^{2}
13 1−4T+13T2 1 - 4T + 13T^{2}
17 1−2iT−17T2 1 - 2iT - 17T^{2}
19 1+(2+2i)T+19iT2 1 + (2 + 2i)T + 19iT^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1+(−7+7i)T−29iT2 1 + (-7 + 7i)T - 29iT^{2}
31 1+(4+4i)T+31iT2 1 + (4 + 4i)T + 31iT^{2}
41 1−41T2 1 - 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+(2−2i)T−47iT2 1 + (2 - 2i)T - 47iT^{2}
53 1+(1+i)T+53iT2 1 + (1 + i)T + 53iT^{2}
59 1+(2+2i)T+59iT2 1 + (2 + 2i)T + 59iT^{2}
61 1+(−1−i)T+61iT2 1 + (-1 - i)T + 61iT^{2}
67 1+67iT2 1 + 67iT^{2}
71 1−12T+71T2 1 - 12T + 71T^{2}
73 1+(−5+5i)T−73iT2 1 + (-5 + 5i)T - 73iT^{2}
79 1+(12+12i)T+79iT2 1 + (12 + 12i)T + 79iT^{2}
83 1+(−4−4i)T+83iT2 1 + (-4 - 4i)T + 83iT^{2}
89 1+(−7+7i)T−89iT2 1 + (-7 + 7i)T - 89iT^{2}
97 1−12iT−97T2 1 - 12iT - 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

−11.47718013602246003375742552267, −10.91660126747867709884420843737, −10.01220533968639049782248084203, −8.664288034810394084445726496172, −7.72891915311317816813286844699, −6.55237808437988363901969639400, −5.94187678596989010516857676987, −4.56135296400034752490946096296, −3.34807128733237594577014040058, −2.36705659868046913175965849507, 1.03199153426286123389538234781, 3.34486315504660437022090896820, 4.01037918190529780212636860751, 5.23245744764443935254708726779, 6.41684643507938113959068339258, 7.17433757636585765934880539061, 8.514673371182734911046364075142, 9.281916531279392146854246427040, 10.47355051026309155446020727050, 11.34700688141165288944093319966

Graph of the ZZ-function along the critical line