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Properties

Label 2-704-11.10-c4-0-5
Degree 22
Conductor 704704
Sign 0.0909−0.995i0.0909 - 0.995i
Analytic cond. 72.772472.7724
Root an. cond. 8.530678.53067
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 31·5-s − 54.7i·7-s − 72·9-s + (−11 + 120. i)11-s − 186. i·13-s − 93·15-s − 230. i·17-s − 98.5i·19-s − 164. i·21-s + 277·23-s + 336·25-s − 459·27-s − 1.27e3i·29-s − 1.36e3·31-s + ⋯
L(s)  = 1  + 0.333·3-s − 1.23·5-s − 1.11i·7-s − 0.888·9-s + (−0.0909 + 0.995i)11-s − 1.10i·13-s − 0.413·15-s − 0.795i·17-s − 0.273i·19-s − 0.372i·21-s + 0.523·23-s + 0.537·25-s − 0.629·27-s − 1.51i·29-s − 1.41·31-s + ⋯

Functional equation

Λ(s)=(704s/2 ΓC(s) L(s)=((0.0909−0.995i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0909 - 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(704s/2 ΓC(s+2) L(s)=((0.0909−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0909 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 704704    =    26⋅112^{6} \cdot 11
Sign: 0.0909−0.995i0.0909 - 0.995i
Analytic conductor: 72.772472.7724
Root analytic conductor: 8.530678.53067
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ704(65,⋅)\chi_{704} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 704, ( :2), 0.0909−0.995i)(2,\ 704,\ (\ :2),\ 0.0909 - 0.995i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.41406699650.4140669965
L(12)L(\frac12) ≈\approx 0.41406699650.4140669965
L(3)L(3) 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
11 1+(11−120.i)T 1 + (11 - 120. i)T
good3 1−3T+81T2 1 - 3T + 81T^{2}
5 1+31T+625T2 1 + 31T + 625T^{2}
7 1+54.7iT−2.40e3T2 1 + 54.7iT - 2.40e3T^{2}
13 1+186.iT−2.85e4T2 1 + 186. iT - 2.85e4T^{2}
17 1+230.iT−8.35e4T2 1 + 230. iT - 8.35e4T^{2}
19 1+98.5iT−1.30e5T2 1 + 98.5iT - 1.30e5T^{2}
23 1−277T+2.79e5T2 1 - 277T + 2.79e5T^{2}
29 1+1.27e3iT−7.07e5T2 1 + 1.27e3iT - 7.07e5T^{2}
31 1+1.36e3T+9.23e5T2 1 + 1.36e3T + 9.23e5T^{2}
37 1+167T+1.87e6T2 1 + 167T + 1.87e6T^{2}
41 1−1.06e3iT−2.82e6T2 1 - 1.06e3iT - 2.82e6T^{2}
43 1−1.20e3iT−3.41e6T2 1 - 1.20e3iT - 3.41e6T^{2}
47 1−1.70e3T+4.87e6T2 1 - 1.70e3T + 4.87e6T^{2}
53 1+4.52e3T+7.89e6T2 1 + 4.52e3T + 7.89e6T^{2}
59 1−2.36e3T+1.21e7T2 1 - 2.36e3T + 1.21e7T^{2}
61 1−3.96e3iT−1.38e7T2 1 - 3.96e3iT - 1.38e7T^{2}
67 1−2.80e3T+2.01e7T2 1 - 2.80e3T + 2.01e7T^{2}
71 1−3.39e3T+2.54e7T2 1 - 3.39e3T + 2.54e7T^{2}
73 1−3.31e3iT−2.83e7T2 1 - 3.31e3iT - 2.83e7T^{2}
79 1−6.09e3iT−3.89e7T2 1 - 6.09e3iT - 3.89e7T^{2}
83 1−832.iT−4.74e7T2 1 - 832. iT - 4.74e7T^{2}
89 1+4.67e3T+6.27e7T2 1 + 4.67e3T + 6.27e7T^{2}
97 1−4.24e3T+8.85e7T2 1 - 4.24e3T + 8.85e7T^{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

−10.06853075379051633131810298118, −9.189871428937191474857315233752, −7.995560491095974618971981760777, −7.67701933808390728888563043843, −6.84810277583866490133958792451, −5.42478531134846722144719256251, −4.39515106126636700288914919212, −3.57575103950064562959230794203, −2.58039569036918823535186869760, −0.77856916563066292387795694586, 0.12824625866850383960652776643, 1.89553145880538082450455823708, 3.16457504191411630658085904645, 3.81870389034627952496839244213, 5.18690037220127878714322917626, 6.01716556082621729049306510803, 7.13012360297796934388139903105, 8.175886485379394366901319879178, 8.730528984883917210880206520326, 9.239860900482506636705237142797

Graph of the ZZ-function along the critical line