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Properties

Label 2-650-5.4-c5-0-72
Degree 22
Conductor 650650
Sign −0.447+0.894i-0.447 + 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 + 27.8i·3-s − 16·4-s + 111.·6-s − 240. i·7-s + 64i·8-s − 534.·9-s + 544.·11-s − 446. i·12-s − 169i·13-s − 961.·14-s + 256·16-s + 1.62e3i·17-s + 2.13e3i·18-s + 805.·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.78i·3-s − 0.5·4-s + 1.26·6-s − 1.85i·7-s + 0.353i·8-s − 2.20·9-s + 1.35·11-s − 0.894i·12-s − 0.277i·13-s − 1.31·14-s + 0.250·16-s + 1.36i·17-s + 1.55i·18-s + 0.512·19-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+0.894i-0.447 + 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 0.92633624960.9263362496
L(12)L(\frac12) ≈\approx 0.92633624960.9263362496
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 1−27.8iT−243T2 1 - 27.8iT - 243T^{2}
7 1+240.iT−1.68e4T2 1 + 240. iT - 1.68e4T^{2}
11 1−544.T+1.61e5T2 1 - 544.T + 1.61e5T^{2}
17 1−1.62e3iT−1.41e6T2 1 - 1.62e3iT - 1.41e6T^{2}
19 1−805.T+2.47e6T2 1 - 805.T + 2.47e6T^{2}
23 1+373.iT−6.43e6T2 1 + 373. iT - 6.43e6T^{2}
29 1+1.50e3T+2.05e7T2 1 + 1.50e3T + 2.05e7T^{2}
31 1+2.20e3T+2.86e7T2 1 + 2.20e3T + 2.86e7T^{2}
37 1+1.31e4iT−6.93e7T2 1 + 1.31e4iT - 6.93e7T^{2}
41 1+1.70e4T+1.15e8T2 1 + 1.70e4T + 1.15e8T^{2}
43 1−8.93e3iT−1.47e8T2 1 - 8.93e3iT - 1.47e8T^{2}
47 1−1.57e4iT−2.29e8T2 1 - 1.57e4iT - 2.29e8T^{2}
53 1+4.03e4iT−4.18e8T2 1 + 4.03e4iT - 4.18e8T^{2}
59 1−4.75e4T+7.14e8T2 1 - 4.75e4T + 7.14e8T^{2}
61 1+3.02e4T+8.44e8T2 1 + 3.02e4T + 8.44e8T^{2}
67 1−3.87e4iT−1.35e9T2 1 - 3.87e4iT - 1.35e9T^{2}
71 1+1.05e4T+1.80e9T2 1 + 1.05e4T + 1.80e9T^{2}
73 1+1.58e3iT−2.07e9T2 1 + 1.58e3iT - 2.07e9T^{2}
79 1−6.19e3T+3.07e9T2 1 - 6.19e3T + 3.07e9T^{2}
83 1+3.78e4iT−3.93e9T2 1 + 3.78e4iT - 3.93e9T^{2}
89 1+4.91e4T+5.58e9T2 1 + 4.91e4T + 5.58e9T^{2}
97 1+1.56e4iT−8.58e9T2 1 + 1.56e4iT - 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

−9.830973968996764816148190015197, −9.012597296138563702460289968973, −8.069656389216261919352617882285, −6.76966304433440673883208677645, −5.51958056086285041228189990275, −4.36831362151893778979758398702, −3.90278186515915168077896038555, −3.34943704491040442423586923119, −1.46594533857512208867919898159, −0.21259995299307858183629994526, 1.15909781236412628448793709589, 2.13167017886666208782088399736, 3.19164173649020223294758534487, 5.08289332738105736889361941602, 5.85021607951043054875512808582, 6.63239496791459946110249064037, 7.20127951100267049300909356070, 8.280629677325761355238264163164, 8.896905239909599215710969269402, 9.524956311768957242913475234189

Graph of the ZZ-function along the critical line