Properties

Label 2-280-8.5-c1-0-2
Degree $2$
Conductor $280$
Sign $-0.937 + 0.348i$
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  + (0.167 + 1.40i)2-s + 2.41i·3-s + (−1.94 + 0.470i)4-s + i·5-s + (−3.39 + 0.404i)6-s − 7-s + (−0.985 − 2.65i)8-s − 2.82·9-s + (−1.40 + 0.167i)10-s − 2.49i·11-s + (−1.13 − 4.69i)12-s + 6.97i·13-s + (−0.167 − 1.40i)14-s − 2.41·15-s + (3.55 − 1.82i)16-s + 4.03·17-s + ⋯
L(s)  = 1  + (0.118 + 0.992i)2-s + 1.39i·3-s + (−0.971 + 0.235i)4-s + 0.447i·5-s + (−1.38 + 0.165i)6-s − 0.377·7-s + (−0.348 − 0.937i)8-s − 0.942·9-s + (−0.444 + 0.0529i)10-s − 0.753i·11-s + (−0.327 − 1.35i)12-s + 1.93i·13-s + (−0.0447 − 0.375i)14-s − 0.623·15-s + (0.889 − 0.457i)16-s + 0.977·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.937 + 0.348i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.937 + 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188169 - 1.04577i\)
\(L(\frac12)\) \(\approx\) \(0.188169 - 1.04577i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.167 - 1.40i)T \)
5 \( 1 - iT \)
7 \( 1 + T \)
good3 \( 1 - 2.41iT - 3T^{2} \)
11 \( 1 + 2.49iT - 11T^{2} \)
13 \( 1 - 6.97iT - 13T^{2} \)
17 \( 1 - 4.03T + 17T^{2} \)
19 \( 1 + 4.64iT - 19T^{2} \)
23 \( 1 + 2.35T + 23T^{2} \)
29 \( 1 - 9.33iT - 29T^{2} \)
31 \( 1 + 0.315T + 31T^{2} \)
37 \( 1 + 7.30iT - 37T^{2} \)
41 \( 1 + 2.64T + 41T^{2} \)
43 \( 1 - 1.68iT - 43T^{2} \)
47 \( 1 - 4.18T + 47T^{2} \)
53 \( 1 - 9.11iT - 53T^{2} \)
59 \( 1 - 11.7iT - 59T^{2} \)
61 \( 1 + 5.03iT - 61T^{2} \)
67 \( 1 + 5.11iT - 67T^{2} \)
71 \( 1 - 7.89T + 71T^{2} \)
73 \( 1 - 9.89T + 73T^{2} \)
79 \( 1 - 2.99T + 79T^{2} \)
83 \( 1 + 4.11iT - 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 - 17.7T + 97T^{2} \)
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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

−12.40631229419255699688736476723, −11.23478093344205288030169458212, −10.31124569998540946681221018542, −9.249907363364219972900548909257, −8.925636646107868657138526471982, −7.36725370882089239812187835998, −6.40794153546061316515575342830, −5.28358459059158632340999585719, −4.22970037971459393661806953558, −3.33386684072321117868862346840, 0.822670326486147477649656091538, 2.20692449009271089806317985302, 3.58167701314939812941250014243, 5.24350371540809625595153757818, 6.19280512107900000351368381566, 7.83497473567751733946190790064, 8.143420939619945551694450649427, 9.776097802878067840181601709135, 10.28770695815557750404206822322, 11.81740020678353690434733012099

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