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Properties

Label 2-650-5.4-c5-0-17
Degree 22
Conductor 650650
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 104.249104.249
Root an. cond. 10.210210.2102
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 − 13.8i·3-s − 16·4-s − 55.2·6-s + 213. i·7-s + 64i·8-s + 52.2·9-s + 587.·11-s + 220. i·12-s + 169i·13-s + 852.·14-s + 256·16-s + 162. i·17-s − 208. i·18-s + 81.3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.886i·3-s − 0.5·4-s − 0.626·6-s + 1.64i·7-s + 0.353i·8-s + 0.215·9-s + 1.46·11-s + 0.443i·12-s + 0.277i·13-s + 1.16·14-s + 0.250·16-s + 0.136i·17-s − 0.152i·18-s + 0.0517·19-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+5/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{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: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 104.249104.249
Root analytic conductor: 10.210210.2102
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ650(599,⋅)\chi_{650} (599, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :5/2), 0.447−0.894i)(2,\ 650,\ (\ :5/2),\ 0.447 - 0.894i)

Particular Values

L(3)L(3) ≈\approx 1.2700690011.270069001
L(12)L(\frac12) ≈\approx 1.2700690011.270069001
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
13 1−169iT 1 - 169iT
good3 1+13.8iT−243T2 1 + 13.8iT - 243T^{2}
7 1−213.iT−1.68e4T2 1 - 213. iT - 1.68e4T^{2}
11 1−587.T+1.61e5T2 1 - 587.T + 1.61e5T^{2}
17 1−162.iT−1.41e6T2 1 - 162. iT - 1.41e6T^{2}
19 1−81.3T+2.47e6T2 1 - 81.3T + 2.47e6T^{2}
23 1−2.94e3iT−6.43e6T2 1 - 2.94e3iT - 6.43e6T^{2}
29 1+6.25e3T+2.05e7T2 1 + 6.25e3T + 2.05e7T^{2}
31 1+4.03e3T+2.86e7T2 1 + 4.03e3T + 2.86e7T^{2}
37 1−7.61e3iT−6.93e7T2 1 - 7.61e3iT - 6.93e7T^{2}
41 1−958.T+1.15e8T2 1 - 958.T + 1.15e8T^{2}
43 1+169.iT−1.47e8T2 1 + 169. iT - 1.47e8T^{2}
47 1−2.16e4iT−2.29e8T2 1 - 2.16e4iT - 2.29e8T^{2}
53 1+2.47e4iT−4.18e8T2 1 + 2.47e4iT - 4.18e8T^{2}
59 1+4.01e4T+7.14e8T2 1 + 4.01e4T + 7.14e8T^{2}
61 1+1.63e4T+8.44e8T2 1 + 1.63e4T + 8.44e8T^{2}
67 1−1.83e4iT−1.35e9T2 1 - 1.83e4iT - 1.35e9T^{2}
71 1+7.78e4T+1.80e9T2 1 + 7.78e4T + 1.80e9T^{2}
73 1−6.21e4iT−2.07e9T2 1 - 6.21e4iT - 2.07e9T^{2}
79 1−6.05e4T+3.07e9T2 1 - 6.05e4T + 3.07e9T^{2}
83 1−2.65e3iT−3.93e9T2 1 - 2.65e3iT - 3.93e9T^{2}
89 1+8.22e3T+5.58e9T2 1 + 8.22e3T + 5.58e9T^{2}
97 1+5.38e4iT−8.58e9T2 1 + 5.38e4iT - 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

−9.614499714368035077133828830585, −9.261680168323628678120553833585, −8.352541077279392892496237227050, −7.31032991877676392980267919770, −6.29560717895398549048010405453, −5.53423751981796829250013596871, −4.22913414634812011431053649666, −3.09566887412121521806125259749, −1.86574969004136478667268614338, −1.40786269689817442154076074892, 0.27401470461281120399157178104, 1.42617728506997653133270024657, 3.58962519728403182276347230552, 4.07270718919929608224409319568, 4.85068011196085016120713452229, 6.14316714076633762036357705322, 7.07020564204057022138702898781, 7.62404060215958156092268949219, 8.976310899044234277287898423790, 9.492523325373256196950066681003

Graph of the ZZ-function along the critical line