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Properties

Label 2-3872-11.2-c0-0-2
Degree 22
Conductor 38723872
Sign 0.396−0.917i0.396 - 0.917i
Analytic cond. 1.932371.93237
Root an. cond. 1.390101.39010
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)9-s + (1.34 + 0.437i)13-s + (−1.34 + 0.437i)17-s + (0.809 − 0.587i)25-s + (−0.831 + 1.14i)29-s + (1.61 + 1.17i)37-s + (−0.831 − 1.14i)41-s + (0.309 + 0.951i)49-s + (−0.618 + 1.90i)53-s + (−1.34 + 0.437i)61-s + (0.831 − 1.14i)73-s + (−0.809 − 0.587i)81-s + 2·89-s + (1.34 + 0.437i)101-s + 1.41i·109-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)9-s + (1.34 + 0.437i)13-s + (−1.34 + 0.437i)17-s + (0.809 − 0.587i)25-s + (−0.831 + 1.14i)29-s + (1.61 + 1.17i)37-s + (−0.831 − 1.14i)41-s + (0.309 + 0.951i)49-s + (−0.618 + 1.90i)53-s + (−1.34 + 0.437i)61-s + (0.831 − 1.14i)73-s + (−0.809 − 0.587i)81-s + 2·89-s + (1.34 + 0.437i)101-s + 1.41i·109-s + ⋯

Functional equation

Λ(s)=(3872s/2 ΓC(s) L(s)=((0.396−0.917i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3872s/2 ΓC(s) L(s)=((0.396−0.917i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38723872    =    25⋅1122^{5} \cdot 11^{2}
Sign: 0.396−0.917i0.396 - 0.917i
Analytic conductor: 1.932371.93237
Root analytic conductor: 1.390101.39010
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3872(3137,⋅)\chi_{3872} (3137, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3872, ( :0), 0.396−0.917i)(2,\ 3872,\ (\ :0),\ 0.396 - 0.917i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1811430371.181143037
L(12)L(\frac12) ≈\approx 1.1811430371.181143037
L(1)L(1) 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 1
11 1 1
good3 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
5 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
7 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
13 1+(−1.34−0.437i)T+(0.809+0.587i)T2 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2}
17 1+(1.34−0.437i)T+(0.809−0.587i)T2 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2}
19 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
23 1+T2 1 + T^{2}
29 1+(0.831−1.14i)T+(−0.309−0.951i)T2 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2}
31 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
37 1+(−1.61−1.17i)T+(0.309+0.951i)T2 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2}
41 1+(0.831+1.14i)T+(−0.309+0.951i)T2 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2}
43 1−T2 1 - T^{2}
47 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
53 1+(0.618−1.90i)T+(−0.809−0.587i)T2 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2}
59 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
61 1+(1.34−0.437i)T+(0.809−0.587i)T2 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2}
67 1+T2 1 + T^{2}
71 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
73 1+(−0.831+1.14i)T+(−0.309−0.951i)T2 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2}
79 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
83 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
89 1−2T+T2 1 - 2T + T^{2}
97 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)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

−8.887780150439104764072802091108, −8.113618414792452379323727301467, −7.39586780128980637741165760825, −6.43308909227761896426253641132, −6.00906011236270065415402632883, −4.90954706395442221425906888917, −4.34400123263516743458687488414, −3.34233182067619751519590170843, −2.36303670167254789371035265555, −1.40533591163448978259427146473, 0.70173201103067781215567915997, 2.03397964865895315910821092875, 3.14439973003531019032530336506, 3.81868602102900334185343000195, 4.69049757274454529232047121730, 5.69732314887534612358386864154, 6.32222397794052967917380300566, 6.90190445064745387079403033776, 7.906863579373144942463864031026, 8.568430763925816910904685229750

Graph of the ZZ-function along the critical line