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Properties

Label 2-350-5.4-c5-0-0
Degree 22
Conductor 350350
Sign −0.894−0.447i-0.894 - 0.447i
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 + 2.52i·3-s − 16·4-s + 10.1·6-s + 49i·7-s + 64i·8-s + 236.·9-s − 267.·11-s − 40.4i·12-s + 896. i·13-s + 196·14-s + 256·16-s − 61.1i·17-s − 946. i·18-s − 1.62e3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.162i·3-s − 0.5·4-s + 0.114·6-s + 0.377i·7-s + 0.353i·8-s + 0.973·9-s − 0.667·11-s − 0.0811i·12-s + 1.47i·13-s + 0.267·14-s + 0.250·16-s − 0.0513i·17-s − 0.688i·18-s − 1.03·19-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: −0.894−0.447i-0.894 - 0.447i
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), −0.894−0.447i)(2,\ 350,\ (\ :5/2),\ -0.894 - 0.447i)

Particular Values

L(3)L(3) ≈\approx 0.032397912600.03239791260
L(12)L(\frac12) ≈\approx 0.032397912600.03239791260
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 1−2.52iT−243T2 1 - 2.52iT - 243T^{2}
11 1+267.T+1.61e5T2 1 + 267.T + 1.61e5T^{2}
13 1−896.iT−3.71e5T2 1 - 896. iT - 3.71e5T^{2}
17 1+61.1iT−1.41e6T2 1 + 61.1iT - 1.41e6T^{2}
19 1+1.62e3T+2.47e6T2 1 + 1.62e3T + 2.47e6T^{2}
23 1+4.28e3iT−6.43e6T2 1 + 4.28e3iT - 6.43e6T^{2}
29 1−7.42e3T+2.05e7T2 1 - 7.42e3T + 2.05e7T^{2}
31 1+8.93e3T+2.86e7T2 1 + 8.93e3T + 2.86e7T^{2}
37 1−640.iT−6.93e7T2 1 - 640. iT - 6.93e7T^{2}
41 1−3.87e3T+1.15e8T2 1 - 3.87e3T + 1.15e8T^{2}
43 1+1.97e4iT−1.47e8T2 1 + 1.97e4iT - 1.47e8T^{2}
47 1+2.06e3iT−2.29e8T2 1 + 2.06e3iT - 2.29e8T^{2}
53 1−1.97e4iT−4.18e8T2 1 - 1.97e4iT - 4.18e8T^{2}
59 1+4.64e4T+7.14e8T2 1 + 4.64e4T + 7.14e8T^{2}
61 1+5.39e4T+8.44e8T2 1 + 5.39e4T + 8.44e8T^{2}
67 1−4.46e4iT−1.35e9T2 1 - 4.46e4iT - 1.35e9T^{2}
71 1+5.05e4T+1.80e9T2 1 + 5.05e4T + 1.80e9T^{2}
73 1+2.40e4iT−2.07e9T2 1 + 2.40e4iT - 2.07e9T^{2}
79 1+1.99e3T+3.07e9T2 1 + 1.99e3T + 3.07e9T^{2}
83 1+5.05e4iT−3.93e9T2 1 + 5.05e4iT - 3.93e9T^{2}
89 1+6.03e4T+5.58e9T2 1 + 6.03e4T + 5.58e9T^{2}
97 1−1.21e5iT−8.58e9T2 1 - 1.21e5iT - 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

−10.81361636569777055084295381098, −10.37988969964562802465597432019, −9.230367050323480989993433787266, −8.589436791201945474808697789892, −7.26430957855566921337318736143, −6.23704421236898820378142822265, −4.76051638181626058174911386654, −4.13580239737364879224118844548, −2.61122067145111493156955462707, −1.62335900851171546268618521582, 0.008268051112501743719563215198, 1.39181402855681858559124210733, 3.13974886455495503290148152174, 4.40192015873810799369049840029, 5.42269404384669105920751241472, 6.46060282983758481840909138890, 7.58111648398833761636345980237, 7.986539443461249917783059922530, 9.313562648503002847283029563488, 10.24622599385282984823748174269

Graph of the ZZ-function along the critical line