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Properties

Label 2-3724-931.436-c0-0-0
Degree 22
Conductor 37243724
Sign −0.949−0.315i-0.949 - 0.315i
Analytic cond. 1.858511.85851
Root an. cond. 1.363271.36327
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  + (−0.109 + 1.46i)5-s + (−0.900 + 0.433i)7-s + (0.365 + 0.930i)9-s + (0.455 − 1.16i)11-s + (−1.40 + 1.29i)17-s + (−0.5 + 0.866i)19-s + (−0.535 − 0.496i)23-s + (−1.13 − 0.171i)25-s + (−0.535 − 1.36i)35-s + (−1.72 − 0.829i)43-s + (−1.40 + 0.432i)45-s + (1.78 − 0.268i)47-s + (0.623 − 0.781i)49-s + (1.64 + 0.793i)55-s + (1.19 + 0.367i)61-s + ⋯
L(s)  = 1  + (−0.109 + 1.46i)5-s + (−0.900 + 0.433i)7-s + (0.365 + 0.930i)9-s + (0.455 − 1.16i)11-s + (−1.40 + 1.29i)17-s + (−0.5 + 0.866i)19-s + (−0.535 − 0.496i)23-s + (−1.13 − 0.171i)25-s + (−0.535 − 1.36i)35-s + (−1.72 − 0.829i)43-s + (−1.40 + 0.432i)45-s + (1.78 − 0.268i)47-s + (0.623 − 0.781i)49-s + (1.64 + 0.793i)55-s + (1.19 + 0.367i)61-s + ⋯

Functional equation

Λ(s)=(3724s/2 ΓC(s) L(s)=((−0.949−0.315i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3724s/2 ΓC(s) L(s)=((−0.949−0.315i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37243724    =    22⋅72⋅192^{2} \cdot 7^{2} \cdot 19
Sign: −0.949−0.315i-0.949 - 0.315i
Analytic conductor: 1.858511.85851
Root analytic conductor: 1.363271.36327
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3724(3229,⋅)\chi_{3724} (3229, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3724, ( :0), −0.949−0.315i)(2,\ 3724,\ (\ :0),\ -0.949 - 0.315i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.71371072580.7137107258
L(12)L(\frac12) ≈\approx 0.71371072580.7137107258
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
7 1+(0.900−0.433i)T 1 + (0.900 - 0.433i)T
19 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good3 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
5 1+(0.109−1.46i)T+(−0.988−0.149i)T2 1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2}
11 1+(−0.455+1.16i)T+(−0.733−0.680i)T2 1 + (-0.455 + 1.16i)T + (-0.733 - 0.680i)T^{2}
13 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
17 1+(1.40−1.29i)T+(0.0747−0.997i)T2 1 + (1.40 - 1.29i)T + (0.0747 - 0.997i)T^{2}
23 1+(0.535+0.496i)T+(0.0747+0.997i)T2 1 + (0.535 + 0.496i)T + (0.0747 + 0.997i)T^{2}
29 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
41 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
43 1+(1.72+0.829i)T+(0.623+0.781i)T2 1 + (1.72 + 0.829i)T + (0.623 + 0.781i)T^{2}
47 1+(−1.78+0.268i)T+(0.955−0.294i)T2 1 + (-1.78 + 0.268i)T + (0.955 - 0.294i)T^{2}
53 1+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
59 1+(0.988−0.149i)T2 1 + (0.988 - 0.149i)T^{2}
61 1+(−1.19−0.367i)T+(0.826+0.563i)T2 1 + (-1.19 - 0.367i)T + (0.826 + 0.563i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
73 1+(1.97+0.298i)T+(0.955+0.294i)T2 1 + (1.97 + 0.298i)T + (0.955 + 0.294i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(0.914−1.14i)T+(−0.222−0.974i)T2 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2}
89 1+(0.733−0.680i)T2 1 + (0.733 - 0.680i)T^{2}
97 1−T2 1 - 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

−8.733816088153255774520067036561, −8.479519301126154577843604438093, −7.43166160362749635226407274303, −6.65101996479075985949730215520, −6.24936475616732154453383069491, −5.56628044551996701027822282647, −4.14189059143065900667319945570, −3.63160119499145956504063561616, −2.65899609260268344223543869833, −1.94945831837895790343847654678, 0.39244817662955645289995198493, 1.56465631318067239855286659987, 2.78296147793536805615972380305, 4.06008593857096166952233209463, 4.39430118510698688722182990151, 5.17265396210580929283830367715, 6.30526079081185385394328337247, 6.91036443293047457247662042505, 7.45427908966747987567576936973, 8.692561415991109523236308608743

Graph of the ZZ-function along the critical line