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Properties

Label 2-350-5.4-c5-0-36
Degree 22
Conductor 350350
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 56.134356.1343
Root an. cond. 7.492287.49228
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 3i·3-s − 16·4-s + 12·6-s − 49i·7-s − 64i·8-s + 234·9-s + 405·11-s + 48i·12-s − 391i·13-s + 196·14-s + 256·16-s − 999i·17-s + 936i·18-s − 2.34e3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.192i·3-s − 0.5·4-s + 0.136·6-s − 0.377i·7-s − 0.353i·8-s + 0.962·9-s + 1.00·11-s + 0.0962i·12-s − 0.641i·13-s + 0.267·14-s + 0.250·16-s − 0.838i·17-s + 0.680i·18-s − 1.48·19-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+5/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 56.134356.1343
Root analytic conductor: 7.492287.49228
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ350(99,⋅)\chi_{350} (99, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 350, ( :5/2), 0.447+0.894i)(2,\ 350,\ (\ :5/2),\ 0.447 + 0.894i)

Particular Values

L(3)L(3) ≈\approx 1.5514564111.551456411
L(12)L(\frac12) ≈\approx 1.5514564111.551456411
L(72)L(\frac{7}{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 1−4iT 1 - 4iT
5 1 1
7 1+49iT 1 + 49iT
good3 1+3iT−243T2 1 + 3iT - 243T^{2}
11 1−405T+1.61e5T2 1 - 405T + 1.61e5T^{2}
13 1+391iT−3.71e5T2 1 + 391iT - 3.71e5T^{2}
17 1+999iT−1.41e6T2 1 + 999iT - 1.41e6T^{2}
19 1+2.34e3T+2.47e6T2 1 + 2.34e3T + 2.47e6T^{2}
23 1−2.43e3iT−6.43e6T2 1 - 2.43e3iT - 6.43e6T^{2}
29 1+8.25e3T+2.05e7T2 1 + 8.25e3T + 2.05e7T^{2}
31 1−4.01e3T+2.86e7T2 1 - 4.01e3T + 2.86e7T^{2}
37 1−7.04e3iT−6.93e7T2 1 - 7.04e3iT - 6.93e7T^{2}
41 1−3.33e3T+1.15e8T2 1 - 3.33e3T + 1.15e8T^{2}
43 1+2.35e4iT−1.47e8T2 1 + 2.35e4iT - 1.47e8T^{2}
47 1+1.03e4iT−2.29e8T2 1 + 1.03e4iT - 2.29e8T^{2}
53 1−3.08e3iT−4.18e8T2 1 - 3.08e3iT - 4.18e8T^{2}
59 1−1.88e4T+7.14e8T2 1 - 1.88e4T + 7.14e8T^{2}
61 1−2.16e4T+8.44e8T2 1 - 2.16e4T + 8.44e8T^{2}
67 1+5.21e4iT−1.35e9T2 1 + 5.21e4iT - 1.35e9T^{2}
71 1+2.85e4T+1.80e9T2 1 + 2.85e4T + 1.80e9T^{2}
73 1+7.03e4iT−2.07e9T2 1 + 7.03e4iT - 2.07e9T^{2}
79 1+5.88e4T+3.07e9T2 1 + 5.88e4T + 3.07e9T^{2}
83 1−756iT−3.93e9T2 1 - 756iT - 3.93e9T^{2}
89 1+1.35e5T+5.58e9T2 1 + 1.35e5T + 5.58e9T^{2}
97 1+1.10e5iT−8.58e9T2 1 + 1.10e5iT - 8.58e9T^{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

−10.30746709885766710239771793117, −9.500413462088595009244174372353, −8.530502277944782146789754948484, −7.41624060770527472970787346996, −6.84495175321044538955280963000, −5.75913212469939274370912961172, −4.52223876740094904572226963669, −3.61827897192229791230810929295, −1.74760237765255382078927926585, −0.41671397945096206055179035081, 1.29871180177463202266503025227, 2.28889899083210710493360341396, 3.93556482073802739511391746779, 4.41217257539128596325672669537, 5.97913053468422386248293348860, 6.92158222362119014219056018762, 8.307446878714609250873939286612, 9.147879500394021129993785859046, 9.924204737832344596010864951707, 10.87524964580864360048042156699

Graph of the ZZ-function along the critical line