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Properties

Label 2-3751-341.92-c0-0-4
Degree 22
Conductor 37513751
Sign 0.970−0.242i0.970 - 0.242i
Analytic cond. 1.871991.87199
Root an. cond. 1.368201.36820
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 − 1.01i)2-s + (0.618 + 1.90i)4-s + (0.809 − 0.587i)5-s + (0.535 + 1.64i)7-s + (0.535 − 1.64i)8-s + (−0.809 − 0.587i)9-s − 1.73·10-s + (0.927 − 2.85i)14-s + (−0.809 + 0.587i)16-s + (0.535 + 1.64i)18-s + (−0.535 + 1.64i)19-s + (1.61 + 1.17i)20-s + (−2.80 + 2.03i)28-s + (0.809 + 0.587i)31-s + (1.40 + 1.01i)35-s + (0.618 − 1.90i)36-s + ⋯
L(s)  = 1  + (−1.40 − 1.01i)2-s + (0.618 + 1.90i)4-s + (0.809 − 0.587i)5-s + (0.535 + 1.64i)7-s + (0.535 − 1.64i)8-s + (−0.809 − 0.587i)9-s − 1.73·10-s + (0.927 − 2.85i)14-s + (−0.809 + 0.587i)16-s + (0.535 + 1.64i)18-s + (−0.535 + 1.64i)19-s + (1.61 + 1.17i)20-s + (−2.80 + 2.03i)28-s + (0.809 + 0.587i)31-s + (1.40 + 1.01i)35-s + (0.618 − 1.90i)36-s + ⋯

Functional equation

Λ(s)=(3751s/2 ΓC(s) L(s)=((0.970−0.242i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3751s/2 ΓC(s) L(s)=((0.970−0.242i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37513751    =    112⋅3111^{2} \cdot 31
Sign: 0.970−0.242i0.970 - 0.242i
Analytic conductor: 1.871991.87199
Root analytic conductor: 1.368201.36820
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3751(2138,⋅)\chi_{3751} (2138, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3751, ( :0), 0.970−0.242i)(2,\ 3751,\ (\ :0),\ 0.970 - 0.242i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.63459866400.6345986640
L(12)L(\frac12) ≈\approx 0.63459866400.6345986640
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)
bad11 1 1
31 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
good2 1+(1.40+1.01i)T+(0.309+0.951i)T2 1 + (1.40 + 1.01i)T + (0.309 + 0.951i)T^{2}
3 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
5 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}
7 1+(−0.535−1.64i)T+(−0.809+0.587i)T2 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2}
13 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
17 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
19 1+(0.535−1.64i)T+(−0.809−0.587i)T2 1 + (0.535 - 1.64i)T + (-0.809 - 0.587i)T^{2}
23 1−T2 1 - T^{2}
29 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
37 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
41 1+(−0.535+1.64i)T+(−0.809−0.587i)T2 1 + (-0.535 + 1.64i)T + (-0.809 - 0.587i)T^{2}
43 1−T2 1 - T^{2}
47 1+(0.618−1.90i)T+(−0.809−0.587i)T2 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2}
53 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
59 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
61 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
67 1−2T+T2 1 - 2T + T^{2}
71 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}
73 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
79 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
83 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.809+0.587i)T+(0.309+0.951i)T2 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)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

−8.936820598108857011038521102311, −8.302588877921286383729202625129, −7.85894084442959952010750319803, −6.42418836423329217658961449576, −5.76444895518419845895684171301, −5.12785392892484194511481306787, −3.71408830843398440911656233467, −2.72280924965996263969254149150, −2.07569922146852978140872121071, −1.29529551624338625621673305677, 0.60784486415046477133004991480, 1.83401187027954727115620945304, 2.84622676423438866821377833543, 4.31502272736247905218838822685, 5.14503307762853046966170796817, 6.06710692494691351950645116476, 6.75136668097471402435084712749, 7.15749124876104216449666753112, 8.039034842801785065152782019965, 8.394626152332640779432923798448

Graph of the ZZ-function along the critical line