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Properties

Label 2-350-5.4-c5-0-19
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 + 23i·3-s − 16·4-s + 92·6-s + 49i·7-s + 64i·8-s − 286·9-s + 555·11-s − 368i·12-s + 241i·13-s + 196·14-s + 256·16-s − 1.49e3i·17-s + 1.14e3i·18-s + 2.03e3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.47i·3-s − 0.5·4-s + 1.04·6-s + 0.377i·7-s + 0.353i·8-s − 1.17·9-s + 1.38·11-s − 0.737i·12-s + 0.395i·13-s + 0.267·14-s + 0.250·16-s − 1.25i·17-s + 0.832i·18-s + 1.29·19-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 2.2108282362.210828236
L(12)L(\frac12) ≈\approx 2.2108282362.210828236
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 1−23iT−243T2 1 - 23iT - 243T^{2}
11 1−555T+1.61e5T2 1 - 555T + 1.61e5T^{2}
13 1−241iT−3.71e5T2 1 - 241iT - 3.71e5T^{2}
17 1+1.49e3iT−1.41e6T2 1 + 1.49e3iT - 1.41e6T^{2}
19 1−2.03e3T+2.47e6T2 1 - 2.03e3T + 2.47e6T^{2}
23 1−1.23e3iT−6.43e6T2 1 - 1.23e3iT - 6.43e6T^{2}
29 1−5.00e3T+2.05e7T2 1 - 5.00e3T + 2.05e7T^{2}
31 1−5.69e3T+2.86e7T2 1 - 5.69e3T + 2.86e7T^{2}
37 1+5.60e3iT−6.93e7T2 1 + 5.60e3iT - 6.93e7T^{2}
41 1+2.42e3T+1.15e8T2 1 + 2.42e3T + 1.15e8T^{2}
43 1+602iT−1.47e8T2 1 + 602iT - 1.47e8T^{2}
47 1+2.31e4iT−2.29e8T2 1 + 2.31e4iT - 2.29e8T^{2}
53 1−2.52e4iT−4.18e8T2 1 - 2.52e4iT - 4.18e8T^{2}
59 1+5.72e3T+7.14e8T2 1 + 5.72e3T + 7.14e8T^{2}
61 1+3.61e4T+8.44e8T2 1 + 3.61e4T + 8.44e8T^{2}
67 1−6.61e4iT−1.35e9T2 1 - 6.61e4iT - 1.35e9T^{2}
71 1−1.60e4T+1.80e9T2 1 - 1.60e4T + 1.80e9T^{2}
73 1−8.04e4iT−2.07e9T2 1 - 8.04e4iT - 2.07e9T^{2}
79 1−6.41e4T+3.07e9T2 1 - 6.41e4T + 3.07e9T^{2}
83 1−1.06e5iT−3.93e9T2 1 - 1.06e5iT - 3.93e9T^{2}
89 1−7.16e4T+5.58e9T2 1 - 7.16e4T + 5.58e9T^{2}
97 1−1.51e5iT−8.58e9T2 1 - 1.51e5iT - 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

−10.79436551992783854040129524485, −9.665584866555970450980743718289, −9.456745655161154827255724815166, −8.531504222638364019542957352750, −6.96674019568112547110125440797, −5.53427091592031866883421927762, −4.63320630788029861859570397367, −3.75633852047851529771488045172, −2.77116584728048343214059821393, −1.06191224531082288781426379242, 0.73413857434771612288531027909, 1.55533415445948253457156174616, 3.28007492211831578059512920654, 4.66975685954181328755070924381, 6.25026670490442218478676980501, 6.49643343210158103303950850906, 7.63559629086931441032039897191, 8.221087832401596413136055544246, 9.301828437467604673834132144747, 10.46105753580863299980992283273

Graph of the ZZ-function along the critical line