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Properties

Label 2-374-11.4-c1-0-9
Degree 22
Conductor 374374
Sign 0.530+0.847i0.530 + 0.847i
Analytic cond. 2.986402.98640
Root an. cond. 1.728121.72812
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  + (−0.809 − 0.587i)2-s + (0.429 − 1.32i)3-s + (0.309 + 0.951i)4-s + (1.85 − 1.34i)5-s + (−1.12 + 0.817i)6-s + (0.942 + 2.90i)7-s + (0.309 − 0.951i)8-s + (0.863 + 0.627i)9-s − 2.29·10-s + (0.309 − 3.30i)11-s + 1.39·12-s + (5.34 + 3.88i)13-s + (0.942 − 2.90i)14-s + (−0.985 − 3.03i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.248 − 0.763i)3-s + (0.154 + 0.475i)4-s + (0.830 − 0.603i)5-s + (−0.459 + 0.333i)6-s + (0.356 + 1.09i)7-s + (0.109 − 0.336i)8-s + (0.287 + 0.209i)9-s − 0.725·10-s + (0.0931 − 0.995i)11-s + 0.401·12-s + (1.48 + 1.07i)13-s + (0.251 − 0.775i)14-s + (−0.254 − 0.783i)15-s + (−0.202 + 0.146i)16-s + (−0.196 + 0.142i)17-s + ⋯

Functional equation

Λ(s)=(374s/2 ΓC(s) L(s)=((0.530+0.847i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 374 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(374s/2 ΓC(s+1/2) L(s)=((0.530+0.847i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 374 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 374374    =    2⋅11⋅172 \cdot 11 \cdot 17
Sign: 0.530+0.847i0.530 + 0.847i
Analytic conductor: 2.986402.98640
Root analytic conductor: 1.728121.72812
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ374(103,⋅)\chi_{374} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 374, ( :1/2), 0.530+0.847i)(2,\ 374,\ (\ :1/2),\ 0.530 + 0.847i)

Particular Values

L(1)L(1) ≈\approx 1.22132−0.676769i1.22132 - 0.676769i
L(12)L(\frac12) ≈\approx 1.22132−0.676769i1.22132 - 0.676769i
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+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
11 1+(−0.309+3.30i)T 1 + (-0.309 + 3.30i)T
17 1+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
good3 1+(−0.429+1.32i)T+(−2.42−1.76i)T2 1 + (-0.429 + 1.32i)T + (-2.42 - 1.76i)T^{2}
5 1+(−1.85+1.34i)T+(1.54−4.75i)T2 1 + (-1.85 + 1.34i)T + (1.54 - 4.75i)T^{2}
7 1+(−0.942−2.90i)T+(−5.66+4.11i)T2 1 + (-0.942 - 2.90i)T + (-5.66 + 4.11i)T^{2}
13 1+(−5.34−3.88i)T+(4.01+12.3i)T2 1 + (-5.34 - 3.88i)T + (4.01 + 12.3i)T^{2}
19 1+(−0.0488+0.150i)T+(−15.3−11.1i)T2 1 + (-0.0488 + 0.150i)T + (-15.3 - 11.1i)T^{2}
23 1+7.94T+23T2 1 + 7.94T + 23T^{2}
29 1+(1.93+5.95i)T+(−23.4+17.0i)T2 1 + (1.93 + 5.95i)T + (-23.4 + 17.0i)T^{2}
31 1+(1.11+0.809i)T+(9.57+29.4i)T2 1 + (1.11 + 0.809i)T + (9.57 + 29.4i)T^{2}
37 1+(2.68+8.27i)T+(−29.9+21.7i)T2 1 + (2.68 + 8.27i)T + (-29.9 + 21.7i)T^{2}
41 1+(2.33−7.18i)T+(−33.1−24.0i)T2 1 + (2.33 - 7.18i)T + (-33.1 - 24.0i)T^{2}
43 1−1.25T+43T2 1 - 1.25T + 43T^{2}
47 1+(−1.77+5.47i)T+(−38.0−27.6i)T2 1 + (-1.77 + 5.47i)T + (-38.0 - 27.6i)T^{2}
53 1+(1.87+1.35i)T+(16.3+50.4i)T2 1 + (1.87 + 1.35i)T + (16.3 + 50.4i)T^{2}
59 1+(−0.698−2.14i)T+(−47.7+34.6i)T2 1 + (-0.698 - 2.14i)T + (-47.7 + 34.6i)T^{2}
61 1+(4.61−3.34i)T+(18.8−58.0i)T2 1 + (4.61 - 3.34i)T + (18.8 - 58.0i)T^{2}
67 1+2.53T+67T2 1 + 2.53T + 67T^{2}
71 1+(12.2−8.90i)T+(21.9−67.5i)T2 1 + (12.2 - 8.90i)T + (21.9 - 67.5i)T^{2}
73 1+(−4.23−13.0i)T+(−59.0+42.9i)T2 1 + (-4.23 - 13.0i)T + (-59.0 + 42.9i)T^{2}
79 1+(11.6+8.44i)T+(24.4+75.1i)T2 1 + (11.6 + 8.44i)T + (24.4 + 75.1i)T^{2}
83 1+(−10.5+7.64i)T+(25.6−78.9i)T2 1 + (-10.5 + 7.64i)T + (25.6 - 78.9i)T^{2}
89 1+1.57T+89T2 1 + 1.57T + 89T^{2}
97 1+(−3.17−2.30i)T+(29.9+92.2i)T2 1 + (-3.17 - 2.30i)T + (29.9 + 92.2i)T^{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

−11.41934411854539442105418296292, −10.24893172547183085222906172209, −9.087192501923558972466086714404, −8.672726842439180856451389796668, −7.81663300377865254747217716839, −6.35629220345614973391262096136, −5.69568600243164151895838654328, −4.01648946923665165162246099727, −2.24127718975178100684997857127, −1.49709257577270784392572887495, 1.57708054583562620862334466561, 3.45260113001573471554510767425, 4.56448025169199764498573848118, 5.92081404357743250984557654464, 6.85348150686694695914599985229, 7.79352289765152822140224182906, 8.877363591389766658189718902643, 9.921975213592566942232973522185, 10.36310610407145085075959057133, 10.91118667214868669598432002598

Graph of the ZZ-function along the critical line