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Properties

Label 2-630-5.4-c5-0-6
Degree 22
Conductor 630630
Sign −0.178+0.983i-0.178 + 0.983i
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 + (−10 + 55i)5-s + 49i·7-s − 64i·8-s + (−220 − 40i)10-s + 175·11-s − 999i·13-s − 196·14-s + 256·16-s + 1.83e3i·17-s + 1.30e3·19-s + (160 − 880i)20-s + 700i·22-s + 4.19e3i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.178 + 0.983i)5-s + 0.377i·7-s − 0.353i·8-s + (−0.695 − 0.126i)10-s + 0.436·11-s − 1.63i·13-s − 0.267·14-s + 0.250·16-s + 1.53i·17-s + 0.831·19-s + (0.0894 − 0.491i)20-s + 0.308i·22-s + 1.65i·23-s + ⋯

Functional equation

Λ(s)=(630s/2 ΓC(s) L(s)=((−0.178+0.983i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(630s/2 ΓC(s+5/2) L(s)=((−0.178+0.983i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 630630    =    2⋅32⋅5⋅72 \cdot 3^{2} \cdot 5 \cdot 7
Sign: −0.178+0.983i-0.178 + 0.983i
Analytic conductor: 101.041101.041
Root analytic conductor: 10.051910.0519
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ630(379,⋅)\chi_{630} (379, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 630, ( :5/2), −0.178+0.983i)(2,\ 630,\ (\ :5/2),\ -0.178 + 0.983i)

Particular Values

L(3)L(3) ≈\approx 0.37913605210.3791360521
L(12)L(\frac12) ≈\approx 0.37913605210.3791360521
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+(10−55i)T 1 + (10 - 55i)T
7 1−49iT 1 - 49iT
good11 1−175T+1.61e5T2 1 - 175T + 1.61e5T^{2}
13 1+999iT−3.71e5T2 1 + 999iT - 3.71e5T^{2}
17 1−1.83e3iT−1.41e6T2 1 - 1.83e3iT - 1.41e6T^{2}
19 1−1.30e3T+2.47e6T2 1 - 1.30e3T + 2.47e6T^{2}
23 1−4.19e3iT−6.43e6T2 1 - 4.19e3iT - 6.43e6T^{2}
29 1+981T+2.05e7T2 1 + 981T + 2.05e7T^{2}
31 1+4.51e3T+2.86e7T2 1 + 4.51e3T + 2.86e7T^{2}
37 1−578iT−6.93e7T2 1 - 578iT - 6.93e7T^{2}
41 1+1.95e4T+1.15e8T2 1 + 1.95e4T + 1.15e8T^{2}
43 1−1.02e4iT−1.47e8T2 1 - 1.02e4iT - 1.47e8T^{2}
47 1+2.56e4iT−2.29e8T2 1 + 2.56e4iT - 2.29e8T^{2}
53 1−2.98e4iT−4.18e8T2 1 - 2.98e4iT - 4.18e8T^{2}
59 1−1.35e3T+7.14e8T2 1 - 1.35e3T + 7.14e8T^{2}
61 1+1.30e4T+8.44e8T2 1 + 1.30e4T + 8.44e8T^{2}
67 1+3.30e4iT−1.35e9T2 1 + 3.30e4iT - 1.35e9T^{2}
71 1−2.19e4T+1.80e9T2 1 - 2.19e4T + 1.80e9T^{2}
73 1−8.37e4iT−2.07e9T2 1 - 8.37e4iT - 2.07e9T^{2}
79 1−6.41e3T+3.07e9T2 1 - 6.41e3T + 3.07e9T^{2}
83 1+7.32e3iT−3.93e9T2 1 + 7.32e3iT - 3.93e9T^{2}
89 1+8.08e4T+5.58e9T2 1 + 8.08e4T + 5.58e9T^{2}
97 1+7.85e4iT−8.58e9T2 1 + 7.85e4iT - 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

−10.32649329937126826631450611010, −9.622367024040758668078703613659, −8.450745758242235071371774879490, −7.75625838617911555206903034974, −6.96780557869018409829727818627, −5.91348945131677205917222265841, −5.38017176341056693083815532636, −3.77833282633999984408914891542, −3.13729642734224738048804065976, −1.54670497090417014251526066946, 0.088866571507929971784595580389, 1.09974638512053633127854037312, 2.13435792778833599794957645534, 3.54721993927074358501300979625, 4.50300841856144636563315184418, 5.11191410971818362560157959415, 6.56698017196206863293800743598, 7.47860949992633766863685573132, 8.675682570444741274093642801523, 9.230230130493563012245875237305

Graph of the ZZ-function along the critical line