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Properties

Label 2-280-5.4-c5-0-13
Degree 22
Conductor 280280
Sign −0.469−0.883i-0.469 - 0.883i
Analytic cond. 44.907444.9074
Root an. cond. 6.701306.70130
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  + 26.0i·3-s + (−26.2 − 49.3i)5-s + 49i·7-s − 436.·9-s + 711.·11-s − 1.00e3i·13-s + (1.28e3 − 683. i)15-s + 1.70e3i·17-s + 1.91e3·19-s − 1.27e3·21-s + 3.81e3i·23-s + (−1.74e3 + 2.58e3i)25-s − 5.05e3i·27-s + 3.52e3·29-s − 4.04e3·31-s + ⋯
L(s)  = 1  + 1.67i·3-s + (−0.469 − 0.883i)5-s + 0.377i·7-s − 1.79·9-s + 1.77·11-s − 1.65i·13-s + (1.47 − 0.784i)15-s + 1.43i·17-s + 1.21·19-s − 0.632·21-s + 1.50i·23-s + (−0.559 + 0.828i)25-s − 1.33i·27-s + 0.777·29-s − 0.755·31-s + ⋯

Functional equation

Λ(s)=(280s/2 ΓC(s) L(s)=((−0.469−0.883i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(280s/2 ΓC(s+5/2) L(s)=((−0.469−0.883i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 280280    =    23⋅5⋅72^{3} \cdot 5 \cdot 7
Sign: −0.469−0.883i-0.469 - 0.883i
Analytic conductor: 44.907444.9074
Root analytic conductor: 6.701306.70130
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ280(169,⋅)\chi_{280} (169, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 280, ( :5/2), −0.469−0.883i)(2,\ 280,\ (\ :5/2),\ -0.469 - 0.883i)

Particular Values

L(3)L(3) ≈\approx 1.8403415391.840341539
L(12)L(\frac12) ≈\approx 1.8403415391.840341539
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 1
5 1+(26.2+49.3i)T 1 + (26.2 + 49.3i)T
7 1−49iT 1 - 49iT
good3 1−26.0iT−243T2 1 - 26.0iT - 243T^{2}
11 1−711.T+1.61e5T2 1 - 711.T + 1.61e5T^{2}
13 1+1.00e3iT−3.71e5T2 1 + 1.00e3iT - 3.71e5T^{2}
17 1−1.70e3iT−1.41e6T2 1 - 1.70e3iT - 1.41e6T^{2}
19 1−1.91e3T+2.47e6T2 1 - 1.91e3T + 2.47e6T^{2}
23 1−3.81e3iT−6.43e6T2 1 - 3.81e3iT - 6.43e6T^{2}
29 1−3.52e3T+2.05e7T2 1 - 3.52e3T + 2.05e7T^{2}
31 1+4.04e3T+2.86e7T2 1 + 4.04e3T + 2.86e7T^{2}
37 1+6.34e3iT−6.93e7T2 1 + 6.34e3iT - 6.93e7T^{2}
41 1−3.36e3T+1.15e8T2 1 - 3.36e3T + 1.15e8T^{2}
43 1−7.74iT−1.47e8T2 1 - 7.74iT - 1.47e8T^{2}
47 1−7.47e3iT−2.29e8T2 1 - 7.47e3iT - 2.29e8T^{2}
53 1+7.61e3iT−4.18e8T2 1 + 7.61e3iT - 4.18e8T^{2}
59 1−3.07e4T+7.14e8T2 1 - 3.07e4T + 7.14e8T^{2}
61 1+3.42e4T+8.44e8T2 1 + 3.42e4T + 8.44e8T^{2}
67 1−4.23e4iT−1.35e9T2 1 - 4.23e4iT - 1.35e9T^{2}
71 1+2.10e4T+1.80e9T2 1 + 2.10e4T + 1.80e9T^{2}
73 1+5.38e3iT−2.07e9T2 1 + 5.38e3iT - 2.07e9T^{2}
79 1+5.47e3T+3.07e9T2 1 + 5.47e3T + 3.07e9T^{2}
83 1−3.12e4iT−3.93e9T2 1 - 3.12e4iT - 3.93e9T^{2}
89 1−2.53e4T+5.58e9T2 1 - 2.53e4T + 5.58e9T^{2}
97 1−6.00e4iT−8.58e9T2 1 - 6.00e4iT - 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

−11.28659188466008338906042524158, −10.23874460946757221235035571001, −9.369754120071252802182616110300, −8.796147755840132576765488339071, −7.75862417096132051732118941032, −5.87217742702247560609457062561, −5.17952774377948260449694506914, −3.95514592415180164011899544399, −3.42138392079361631356403638299, −1.13975789508088119970262117512, 0.60287525681054035563628802796, 1.72071687333095428338303404125, 2.99206825645812053212956633981, 4.36731053144520171152802134966, 6.29969205247079320135844072683, 6.90575843564027903390845581692, 7.31958788421109641159675679113, 8.621343086603898869133834729070, 9.599879082676025983170495023519, 11.18759599353112629689627682442

Graph of the ZZ-function along the critical line