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Properties

Label 2-370-185.103-c1-0-6
Degree 22
Conductor 370370
Sign 0.946+0.322i0.946 + 0.322i
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 + (−0.241 − 0.900i)3-s + (−0.499 + 0.866i)4-s + (2.17 + 0.519i)5-s + (−0.658 + 0.658i)6-s + (0.596 + 2.22i)7-s + 0.999·8-s + (1.84 − 1.06i)9-s + (−0.637 − 2.14i)10-s + 4.55i·11-s + (0.900 + 0.241i)12-s + (−2.24 + 3.88i)13-s + (1.62 − 1.62i)14-s + (−0.0572 − 2.08i)15-s + (−0.5 − 0.866i)16-s + (0.121 − 0.0699i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.139 − 0.519i)3-s + (−0.249 + 0.433i)4-s + (0.972 + 0.232i)5-s + (−0.269 + 0.269i)6-s + (0.225 + 0.841i)7-s + 0.353·8-s + (0.615 − 0.355i)9-s + (−0.201 − 0.677i)10-s + 1.37i·11-s + (0.259 + 0.0696i)12-s + (−0.622 + 1.07i)13-s + (0.435 − 0.435i)14-s + (−0.0147 − 0.537i)15-s + (−0.125 − 0.216i)16-s + (0.0293 − 0.0169i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.946+0.322i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.946+0.322i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.946+0.322i0.946 + 0.322i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(103,⋅)\chi_{370} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.946+0.322i)(2,\ 370,\ (\ :1/2),\ 0.946 + 0.322i)

Particular Values

L(1)L(1) ≈\approx 1.28191−0.212670i1.28191 - 0.212670i
L(12)L(\frac12) ≈\approx 1.28191−0.212670i1.28191 - 0.212670i
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+(−2.17−0.519i)T 1 + (-2.17 - 0.519i)T
37 1+(2.51+5.53i)T 1 + (2.51 + 5.53i)T
good3 1+(0.241+0.900i)T+(−2.59+1.5i)T2 1 + (0.241 + 0.900i)T + (-2.59 + 1.5i)T^{2}
7 1+(−0.596−2.22i)T+(−6.06+3.5i)T2 1 + (-0.596 - 2.22i)T + (-6.06 + 3.5i)T^{2}
11 1−4.55iT−11T2 1 - 4.55iT - 11T^{2}
13 1+(2.24−3.88i)T+(−6.5−11.2i)T2 1 + (2.24 - 3.88i)T + (-6.5 - 11.2i)T^{2}
17 1+(−0.121+0.0699i)T+(8.5−14.7i)T2 1 + (-0.121 + 0.0699i)T + (8.5 - 14.7i)T^{2}
19 1+(−4.17+1.11i)T+(16.4−9.5i)T2 1 + (-4.17 + 1.11i)T + (16.4 - 9.5i)T^{2}
23 1−0.446T+23T2 1 - 0.446T + 23T^{2}
29 1+(−4.72+4.72i)T−29iT2 1 + (-4.72 + 4.72i)T - 29iT^{2}
31 1+(4.75+4.75i)T+31iT2 1 + (4.75 + 4.75i)T + 31iT^{2}
41 1+(−1.14−0.659i)T+(20.5+35.5i)T2 1 + (-1.14 - 0.659i)T + (20.5 + 35.5i)T^{2}
43 1+9.82T+43T2 1 + 9.82T + 43T^{2}
47 1+(−4.73+4.73i)T−47iT2 1 + (-4.73 + 4.73i)T - 47iT^{2}
53 1+(0.694−2.59i)T+(−45.8−26.5i)T2 1 + (0.694 - 2.59i)T + (-45.8 - 26.5i)T^{2}
59 1+(2.78−10.3i)T+(−51.0−29.5i)T2 1 + (2.78 - 10.3i)T + (-51.0 - 29.5i)T^{2}
61 1+(−10.7+2.88i)T+(52.8−30.5i)T2 1 + (-10.7 + 2.88i)T + (52.8 - 30.5i)T^{2}
67 1+(8.53−2.28i)T+(58.0−33.5i)T2 1 + (8.53 - 2.28i)T + (58.0 - 33.5i)T^{2}
71 1+(7.49−12.9i)T+(−35.5−61.4i)T2 1 + (7.49 - 12.9i)T + (-35.5 - 61.4i)T^{2}
73 1+(−3.51+3.51i)T−73iT2 1 + (-3.51 + 3.51i)T - 73iT^{2}
79 1+(−8.56+2.29i)T+(68.4−39.5i)T2 1 + (-8.56 + 2.29i)T + (68.4 - 39.5i)T^{2}
83 1+(−0.647+2.41i)T+(−71.8−41.5i)T2 1 + (-0.647 + 2.41i)T + (-71.8 - 41.5i)T^{2}
89 1+(17.1+4.58i)T+(77.0+44.5i)T2 1 + (17.1 + 4.58i)T + (77.0 + 44.5i)T^{2}
97 1+11.6iT−97T2 1 + 11.6iT - 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

−11.61298215238718135634583418485, −10.17279741327359662512949360756, −9.630988702798691824637141300505, −8.935964792673732921134572024579, −7.42523630346804766148505342898, −6.80181834924241748686430076424, −5.50032122260540275780474743813, −4.32271886074056810325727401985, −2.45678291400891107132686192695, −1.69476201171934995485527908276, 1.19383863532323832498297540449, 3.31039309656927263989363384343, 4.91596063157079365099296687593, 5.47574017364619384850717320524, 6.70948478861976866827801117838, 7.72159650729549086687091000952, 8.669817493296787716749204127620, 9.738956786600275880682425028761, 10.37721713794618902336258157906, 10.95224385634253901195735672682

Graph of the ZZ-function along the critical line