Properties

Label 2-280-35.12-c1-0-10
Degree $2$
Conductor $280$
Sign $-0.157 + 0.987i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.590 − 2.20i)3-s + (1.73 − 1.40i)5-s + (−2.34 − 1.22i)7-s + (−1.90 − 1.10i)9-s + (−0.0644 − 0.111i)11-s + (0.748 + 0.748i)13-s + (−2.07 − 4.65i)15-s + (2.28 + 0.613i)17-s + (−3.81 + 6.59i)19-s + (−4.08 + 4.44i)21-s + (−1.10 − 4.11i)23-s + (1.02 − 4.89i)25-s + (1.28 − 1.28i)27-s − 0.163i·29-s + (8.43 − 4.86i)31-s + ⋯
L(s)  = 1  + (0.340 − 1.27i)3-s + (0.776 − 0.630i)5-s + (−0.886 − 0.462i)7-s + (−0.636 − 0.367i)9-s + (−0.0194 − 0.0336i)11-s + (0.207 + 0.207i)13-s + (−0.536 − 1.20i)15-s + (0.555 + 0.148i)17-s + (−0.874 + 1.51i)19-s + (−0.890 + 0.970i)21-s + (−0.229 − 0.857i)23-s + (0.205 − 0.978i)25-s + (0.246 − 0.246i)27-s − 0.0303i·29-s + (1.51 − 0.874i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.157 + 0.987i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.157 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945581 - 1.10814i\)
\(L(\frac12)\) \(\approx\) \(0.945581 - 1.10814i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.73 + 1.40i)T \)
7 \( 1 + (2.34 + 1.22i)T \)
good3 \( 1 + (-0.590 + 2.20i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.0644 + 0.111i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.748 - 0.748i)T + 13iT^{2} \)
17 \( 1 + (-2.28 - 0.613i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.81 - 6.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.10 + 4.11i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.163iT - 29T^{2} \)
31 \( 1 + (-8.43 + 4.86i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0900 - 0.0241i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 11.9iT - 41T^{2} \)
43 \( 1 + (-2.76 + 2.76i)T - 43iT^{2} \)
47 \( 1 + (-0.597 - 2.23i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-7.32 - 1.96i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.13 - 7.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.5 + 6.64i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.83 - 6.84i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.50T + 71T^{2} \)
73 \( 1 + (-0.483 + 1.80i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.48 - 4.89i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.43 - 8.43i)T + 83iT^{2} \)
89 \( 1 + (3.37 - 5.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.61 + 9.61i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.11263284298570929456520392452, −10.44040716435896192530039282357, −9.738775336002595105094360291355, −8.522047868165479807723125690354, −7.78714841594319234013704224178, −6.48905692346731526632943044566, −6.01471530287465809875265993637, −4.24047241533126765413798354246, −2.55708205476017672909195899664, −1.19712959400371256527270255638, 2.62338728966307810270044280699, 3.55616168790708888411975977672, 4.95981325649096286352816728704, 6.05090168995123394718848742410, 7.07873948827184878451932285255, 8.759193781704226268553021145963, 9.373967095554122489546483248168, 10.20265487451954484828899253731, 10.75930957450259788742666727365, 12.04405041869427657510124079483

Graph of the $Z$-function along the critical line