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Properties

Label 2-370-185.162-c1-0-18
Degree 22
Conductor 370370
Sign −0.998−0.0568i-0.998 - 0.0568i
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  + (−0.866 − 0.5i)2-s + (0.803 − 2.99i)3-s + (0.499 + 0.866i)4-s + (−0.0397 − 2.23i)5-s + (−2.19 + 2.19i)6-s + (−0.886 + 3.30i)7-s − 0.999i·8-s + (−5.75 − 3.32i)9-s + (−1.08 + 1.95i)10-s − 4.03i·11-s + (2.99 − 0.803i)12-s + (−3.29 + 1.90i)13-s + (2.42 − 2.42i)14-s + (−6.73 − 1.67i)15-s + (−0.5 + 0.866i)16-s + (1.42 − 2.47i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.464 − 1.73i)3-s + (0.249 + 0.433i)4-s + (−0.0177 − 0.999i)5-s + (−0.896 + 0.896i)6-s + (−0.335 + 1.25i)7-s − 0.353i·8-s + (−1.91 − 1.10i)9-s + (−0.342 + 0.618i)10-s − 1.21i·11-s + (0.865 − 0.232i)12-s + (−0.914 + 0.527i)13-s + (0.647 − 0.647i)14-s + (−1.73 − 0.433i)15-s + (−0.125 + 0.216i)16-s + (0.346 − 0.600i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((−0.998−0.0568i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0568i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((−0.998−0.0568i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0568i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: −0.998−0.0568i-0.998 - 0.0568i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(347,⋅)\chi_{370} (347, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), −0.998−0.0568i)(2,\ 370,\ (\ :1/2),\ -0.998 - 0.0568i)

Particular Values

L(1)L(1) ≈\approx 0.0256792+0.903373i0.0256792 + 0.903373i
L(12)L(\frac12) ≈\approx 0.0256792+0.903373i0.0256792 + 0.903373i
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
5 1+(0.0397+2.23i)T 1 + (0.0397 + 2.23i)T
37 1+(−5.52−2.54i)T 1 + (-5.52 - 2.54i)T
good3 1+(−0.803+2.99i)T+(−2.59−1.5i)T2 1 + (-0.803 + 2.99i)T + (-2.59 - 1.5i)T^{2}
7 1+(0.886−3.30i)T+(−6.06−3.5i)T2 1 + (0.886 - 3.30i)T + (-6.06 - 3.5i)T^{2}
11 1+4.03iT−11T2 1 + 4.03iT - 11T^{2}
13 1+(3.29−1.90i)T+(6.5−11.2i)T2 1 + (3.29 - 1.90i)T + (6.5 - 11.2i)T^{2}
17 1+(−1.42+2.47i)T+(−8.5−14.7i)T2 1 + (-1.42 + 2.47i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.23+4.61i)T+(−16.4−9.5i)T2 1 + (-1.23 + 4.61i)T + (-16.4 - 9.5i)T^{2}
23 1−5.69iT−23T2 1 - 5.69iT - 23T^{2}
29 1+(−2.30+2.30i)T−29iT2 1 + (-2.30 + 2.30i)T - 29iT^{2}
31 1+(−1.95−1.95i)T+31iT2 1 + (-1.95 - 1.95i)T + 31iT^{2}
41 1+(−4.19+2.42i)T+(20.5−35.5i)T2 1 + (-4.19 + 2.42i)T + (20.5 - 35.5i)T^{2}
43 1+8.10iT−43T2 1 + 8.10iT - 43T^{2}
47 1+(5.47+5.47i)T+47iT2 1 + (5.47 + 5.47i)T + 47iT^{2}
53 1+(−1.82−6.82i)T+(−45.8+26.5i)T2 1 + (-1.82 - 6.82i)T + (-45.8 + 26.5i)T^{2}
59 1+(4.59−1.23i)T+(51.0−29.5i)T2 1 + (4.59 - 1.23i)T + (51.0 - 29.5i)T^{2}
61 1+(−2.89+10.8i)T+(−52.8−30.5i)T2 1 + (-2.89 + 10.8i)T + (-52.8 - 30.5i)T^{2}
67 1+(−5.28−1.41i)T+(58.0+33.5i)T2 1 + (-5.28 - 1.41i)T + (58.0 + 33.5i)T^{2}
71 1+(5.05+8.75i)T+(−35.5+61.4i)T2 1 + (5.05 + 8.75i)T + (-35.5 + 61.4i)T^{2}
73 1+(−7.11−7.11i)T+73iT2 1 + (-7.11 - 7.11i)T + 73iT^{2}
79 1+(−2.88+10.7i)T+(−68.4−39.5i)T2 1 + (-2.88 + 10.7i)T + (-68.4 - 39.5i)T^{2}
83 1+(0.739+2.76i)T+(−71.8+41.5i)T2 1 + (0.739 + 2.76i)T + (-71.8 + 41.5i)T^{2}
89 1+(−3.14−11.7i)T+(−77.0+44.5i)T2 1 + (-3.14 - 11.7i)T + (-77.0 + 44.5i)T^{2}
97 1−10.0T+97T2 1 - 10.0T + 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

−11.46189544736164038972876425959, −9.476916152826930797540415290879, −8.992697390517093035468174701529, −8.234898829064699046129754235565, −7.41117146194210173423161120213, −6.30971889451603803112481111465, −5.30464575393977941898669869359, −3.07315965799864838751772150160, −2.12381037210825388510761882544, −0.69180518356468943798008916151, 2.68513261336204061322058028759, 3.87369325150179591502976041261, 4.75899082066756781044001734818, 6.21264744963204654907233773929, 7.40892336346333059870304430239, 8.098355937011294083817931604201, 9.558822955344094687592774331921, 10.16072022526853185721168758741, 10.29556101280719951970777790102, 11.29298713227088920845123265486

Graph of the ZZ-function along the critical line