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Properties

Label 2-630-21.20-c5-0-16
Degree 22
Conductor 630630
Sign 0.941+0.337i0.941 + 0.337i
Analytic cond. 101.041101.041
Root an. cond. 10.051910.0519
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 − 16·4-s + 25·5-s + (−124. + 34.7i)7-s + 64i·8-s − 100i·10-s − 250. i·11-s + 257. i·13-s + (139. + 499. i)14-s + 256·16-s − 1.65e3·17-s + 117. i·19-s − 400·20-s − 1.00e3·22-s − 3.22e3i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.447·5-s + (−0.963 + 0.268i)7-s + 0.353i·8-s − 0.316i·10-s − 0.625i·11-s + 0.421i·13-s + (0.189 + 0.681i)14-s + 0.250·16-s − 1.38·17-s + 0.0748i·19-s − 0.223·20-s − 0.442·22-s − 1.27i·23-s + ⋯

Functional equation

Λ(s)=(630s/2 ΓC(s) L(s)=((0.941+0.337i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(630s/2 ΓC(s+5/2) L(s)=((0.941+0.337i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 630630    =    2⋅32⋅5⋅72 \cdot 3^{2} \cdot 5 \cdot 7
Sign: 0.941+0.337i0.941 + 0.337i
Analytic conductor: 101.041101.041
Root analytic conductor: 10.051910.0519
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ630(251,⋅)\chi_{630} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 630, ( :5/2), 0.941+0.337i)(2,\ 630,\ (\ :5/2),\ 0.941 + 0.337i)

Particular Values

L(3)L(3) ≈\approx 1.3672540051.367254005
L(12)L(\frac12) ≈\approx 1.3672540051.367254005
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
3 1 1
5 1−25T 1 - 25T
7 1+(124.−34.7i)T 1 + (124. - 34.7i)T
good11 1+250.iT−1.61e5T2 1 + 250. iT - 1.61e5T^{2}
13 1−257.iT−3.71e5T2 1 - 257. iT - 3.71e5T^{2}
17 1+1.65e3T+1.41e6T2 1 + 1.65e3T + 1.41e6T^{2}
19 1−117.iT−2.47e6T2 1 - 117. iT - 2.47e6T^{2}
23 1+3.22e3iT−6.43e6T2 1 + 3.22e3iT - 6.43e6T^{2}
29 1+1.17e3iT−2.05e7T2 1 + 1.17e3iT - 2.05e7T^{2}
31 1−6.32e3iT−2.86e7T2 1 - 6.32e3iT - 2.86e7T^{2}
37 1−8.37e3T+6.93e7T2 1 - 8.37e3T + 6.93e7T^{2}
41 1+1.46e4T+1.15e8T2 1 + 1.46e4T + 1.15e8T^{2}
43 1+1.58e4T+1.47e8T2 1 + 1.58e4T + 1.47e8T^{2}
47 1−1.68e4T+2.29e8T2 1 - 1.68e4T + 2.29e8T^{2}
53 1+2.33e3iT−4.18e8T2 1 + 2.33e3iT - 4.18e8T^{2}
59 1−9.24e3T+7.14e8T2 1 - 9.24e3T + 7.14e8T^{2}
61 1−1.96e4iT−8.44e8T2 1 - 1.96e4iT - 8.44e8T^{2}
67 1−4.67e4T+1.35e9T2 1 - 4.67e4T + 1.35e9T^{2}
71 1−3.64e4iT−1.80e9T2 1 - 3.64e4iT - 1.80e9T^{2}
73 1−2.68e4iT−2.07e9T2 1 - 2.68e4iT - 2.07e9T^{2}
79 1+3.75e4T+3.07e9T2 1 + 3.75e4T + 3.07e9T^{2}
83 1−9.08e4T+3.93e9T2 1 - 9.08e4T + 3.93e9T^{2}
89 1−5.77e4T+5.58e9T2 1 - 5.77e4T + 5.58e9T^{2}
97 1−2.40e4iT−8.58e9T2 1 - 2.40e4iT - 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

−9.886918674471704861385455099489, −8.941463385584470584532865873618, −8.479196659409785845044774892895, −6.85447315170971087283826960903, −6.25322849744826378424136095639, −5.09775822796191360147416774042, −4.01120916519832113066006529962, −2.92455960859360907825514124917, −2.07633582839895236067100154676, −0.64809336888673072532399766695, 0.45699149695329614440894755525, 2.02673929104209399246744308882, 3.34764216249412887105583332035, 4.42878530506830058370616263481, 5.47935412498142404858373939679, 6.40952565023545555507335236067, 7.04799819349890571031381609182, 7.999017251947045488650160390852, 9.132649324009379438817488194624, 9.666080009552955243084237801340

Graph of the ZZ-function along the critical line