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Properties

Label 2-370-37.26-c1-0-0
Degree 22
Conductor 370370
Sign −0.993−0.115i-0.993 - 0.115i
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  + (−0.5 − 0.866i)2-s + (−1.38 + 2.40i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 2.77·6-s + (−1.65 + 2.86i)7-s + 0.999·8-s + (−2.35 − 4.08i)9-s − 0.999·10-s + 1.52·11-s + (−1.38 − 2.40i)12-s + (−2.09 + 3.62i)13-s + 3.30·14-s + (1.38 + 2.40i)15-s + (−0.5 − 0.866i)16-s + (−3.90 − 6.76i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.802 + 1.38i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + 1.13·6-s + (−0.625 + 1.08i)7-s + 0.353·8-s + (−0.786 − 1.36i)9-s − 0.316·10-s + 0.461·11-s + (−0.401 − 0.694i)12-s + (−0.581 + 1.00i)13-s + 0.884·14-s + (0.358 + 0.621i)15-s + (−0.125 − 0.216i)16-s + (−0.946 − 1.63i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((−0.993−0.115i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((−0.993−0.115i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: −0.993−0.115i-0.993 - 0.115i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(211,⋅)\chi_{370} (211, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), −0.993−0.115i)(2,\ 370,\ (\ :1/2),\ -0.993 - 0.115i)

Particular Values

L(1)L(1) ≈\approx 0.0164852+0.285005i0.0164852 + 0.285005i
L(12)L(\frac12) ≈\approx 0.0164852+0.285005i0.0164852 + 0.285005i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
37 1+(2.87−5.36i)T 1 + (2.87 - 5.36i)T
good3 1+(1.38−2.40i)T+(−1.5−2.59i)T2 1 + (1.38 - 2.40i)T + (-1.5 - 2.59i)T^{2}
7 1+(1.65−2.86i)T+(−3.5−6.06i)T2 1 + (1.65 - 2.86i)T + (-3.5 - 6.06i)T^{2}
11 1−1.52T+11T2 1 - 1.52T + 11T^{2}
13 1+(2.09−3.62i)T+(−6.5−11.2i)T2 1 + (2.09 - 3.62i)T + (-6.5 - 11.2i)T^{2}
17 1+(3.90+6.76i)T+(−8.5+14.7i)T2 1 + (3.90 + 6.76i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.110+0.191i)T+(−9.5−16.4i)T2 1 + (-0.110 + 0.191i)T + (-9.5 - 16.4i)T^{2}
23 1+3.52T+23T2 1 + 3.52T + 23T^{2}
29 1+6.49T+29T2 1 + 6.49T + 29T^{2}
31 1+4.41T+31T2 1 + 4.41T + 31T^{2}
41 1+(0.610−1.05i)T+(−20.5−35.5i)T2 1 + (0.610 - 1.05i)T + (-20.5 - 35.5i)T^{2}
43 1+0.162T+43T2 1 + 0.162T + 43T^{2}
47 1−11.9T+47T2 1 - 11.9T + 47T^{2}
53 1+(−1.98−3.43i)T+(−26.5+45.8i)T2 1 + (-1.98 - 3.43i)T + (-26.5 + 45.8i)T^{2}
59 1+(0.543+0.940i)T+(−29.5+51.0i)T2 1 + (0.543 + 0.940i)T + (-29.5 + 51.0i)T^{2}
61 1+(−7.24+12.5i)T+(−30.5−52.8i)T2 1 + (-7.24 + 12.5i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.83−6.64i)T+(−33.5−58.0i)T2 1 + (3.83 - 6.64i)T + (-33.5 - 58.0i)T^{2}
71 1+(7.08−12.2i)T+(−35.5−61.4i)T2 1 + (7.08 - 12.2i)T + (-35.5 - 61.4i)T^{2}
73 1+7.80T+73T2 1 + 7.80T + 73T^{2}
79 1+(2.52−4.38i)T+(−39.5−68.4i)T2 1 + (2.52 - 4.38i)T + (-39.5 - 68.4i)T^{2}
83 1+(−4.71−8.17i)T+(−41.5+71.8i)T2 1 + (-4.71 - 8.17i)T + (-41.5 + 71.8i)T^{2}
89 1+(−8.66−15.0i)T+(−44.5+77.0i)T2 1 + (-8.66 - 15.0i)T + (-44.5 + 77.0i)T^{2}
97 1−8.38T+97T2 1 - 8.38T + 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

−11.73670034856786985463734047407, −10.98102386334011149549404219809, −9.774787669171032875843699707645, −9.373769534920697892430461154065, −8.837859671579324453453144079536, −6.98766700914879932548003221969, −5.75348154401639637230193194635, −4.86296514984672649015973965207, −3.89962660111783910739408884010, −2.40628554290662796725594291143, 0.22773485841677310350487891455, 1.84914645517564051397878052737, 3.91203350871080451544902344016, 5.68066756347350282256813898930, 6.27278864180423343487412863745, 7.21557183909617592970986467237, 7.61853496126903900194905281901, 8.933377597502440625277304686583, 10.33268056017700519764503683166, 10.72646302941709621525014874073

Graph of the ZZ-function along the critical line