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Properties

Label 2-280-56.27-c1-0-29
Degree $2$
Conductor $280$
Sign $-0.967 - 0.251i$
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  + (0.244 − 1.39i)2-s − 1.68i·3-s + (−1.88 − 0.679i)4-s − 5-s + (−2.34 − 0.411i)6-s + (−0.695 − 2.55i)7-s + (−1.40 + 2.45i)8-s + 0.163·9-s + (−0.244 + 1.39i)10-s + 1.45·11-s + (−1.14 + 3.16i)12-s − 5.12·13-s + (−3.72 + 0.346i)14-s + 1.68i·15-s + (3.07 + 2.55i)16-s + 0.313i·17-s + ⋯
L(s)  = 1  + (0.172 − 0.984i)2-s − 0.972i·3-s + (−0.940 − 0.339i)4-s − 0.447·5-s + (−0.957 − 0.167i)6-s + (−0.262 − 0.964i)7-s + (−0.497 + 0.867i)8-s + 0.0544·9-s + (−0.0771 + 0.440i)10-s + 0.438·11-s + (−0.330 + 0.914i)12-s − 1.42·13-s + (−0.995 + 0.0925i)14-s + 0.434i·15-s + (0.768 + 0.639i)16-s + 0.0761i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.967 - 0.251i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.967 - 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123104 + 0.963218i\)
\(L(\frac12)\) \(\approx\) \(0.123104 + 0.963218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.244 + 1.39i)T \)
5 \( 1 + T \)
7 \( 1 + (0.695 + 2.55i)T \)
good3 \( 1 + 1.68iT - 3T^{2} \)
11 \( 1 - 1.45T + 11T^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
17 \( 1 - 0.313iT - 17T^{2} \)
19 \( 1 - 0.250iT - 19T^{2} \)
23 \( 1 + 4.27iT - 23T^{2} \)
29 \( 1 - 1.63iT - 29T^{2} \)
31 \( 1 - 8.96T + 31T^{2} \)
37 \( 1 + 3.47iT - 37T^{2} \)
41 \( 1 + 9.88iT - 41T^{2} \)
43 \( 1 + 8.65T + 43T^{2} \)
47 \( 1 - 7.77T + 47T^{2} \)
53 \( 1 + 1.90iT - 53T^{2} \)
59 \( 1 + 7.73iT - 59T^{2} \)
61 \( 1 + 0.415T + 61T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 - 1.50iT - 71T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 - 9.36iT - 79T^{2} \)
83 \( 1 + 3.45iT - 83T^{2} \)
89 \( 1 + 9.12iT - 89T^{2} \)
97 \( 1 - 16.5iT - 97T^{2} \)
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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

−11.59539400442164292595423090566, −10.43274697968230572534863114128, −9.777388555302263860047376128225, −8.445247899019986904492507037803, −7.41600787969671882662316968559, −6.56109921613765911205072805307, −4.85846038364445382775540539965, −3.83450746019172529233247342549, −2.32718820214559491842495830624, −0.72288732398249531708893531661, 3.12911114904549540790372245907, 4.43274890863833177855470897852, 5.15372398766769841576577355021, 6.39852043047291710108838670656, 7.49081467532342029890978179711, 8.555921823549263279694330121330, 9.543178066401440122254072907447, 9.998231016226952900248767993159, 11.65352684730664213937632492033, 12.32165330508194377916717157092

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