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Properties

Label 2-280-8.5-c1-0-17
Degree $2$
Conductor $280$
Sign $0.817 + 0.575i$
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  + (1.34 − 0.443i)2-s + 0.414i·3-s + (1.60 − 1.19i)4-s − i·5-s + (0.183 + 0.556i)6-s − 7-s + (1.62 − 2.31i)8-s + 2.82·9-s + (−0.443 − 1.34i)10-s + 1.54i·11-s + (0.493 + 0.665i)12-s − 1.74i·13-s + (−1.34 + 0.443i)14-s + 0.414·15-s + (1.15 − 3.82i)16-s − 2.63·17-s + ⋯
L(s)  = 1  + (0.949 − 0.313i)2-s + 0.239i·3-s + (0.803 − 0.595i)4-s − 0.447i·5-s + (0.0750 + 0.227i)6-s − 0.377·7-s + (0.575 − 0.817i)8-s + 0.942·9-s + (−0.140 − 0.424i)10-s + 0.465i·11-s + (0.142 + 0.192i)12-s − 0.483i·13-s + (−0.358 + 0.118i)14-s + 0.106·15-s + (0.289 − 0.957i)16-s − 0.640·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.817 + 0.575i$
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: \((2,\ 280,\ (\ :1/2),\ 0.817 + 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10447 - 0.666196i\)
\(L(\frac12)\) \(\approx\) \(2.10447 - 0.666196i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 0.443i)T \)
5 \( 1 + iT \)
7 \( 1 + T \)
good3 \( 1 - 0.414iT - 3T^{2} \)
11 \( 1 - 1.54iT - 11T^{2} \)
13 \( 1 + 1.74iT - 13T^{2} \)
17 \( 1 + 2.63T + 17T^{2} \)
19 \( 1 - 4.11iT - 19T^{2} \)
23 \( 1 + 0.969T + 23T^{2} \)
29 \( 1 - 7.26iT - 29T^{2} \)
31 \( 1 + 6.40T + 31T^{2} \)
37 \( 1 + 2.62iT - 37T^{2} \)
41 \( 1 + 2.11T + 41T^{2} \)
43 \( 1 - 4.40iT - 43T^{2} \)
47 \( 1 + 2.85T + 47T^{2} \)
53 \( 1 + 4.31iT - 53T^{2} \)
59 \( 1 - 12.9iT - 59T^{2} \)
61 \( 1 + 11.6iT - 61T^{2} \)
67 \( 1 - 0.317iT - 67T^{2} \)
71 \( 1 + 9.19T + 71T^{2} \)
73 \( 1 + 7.19T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 1.23iT - 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 - 6.38T + 97T^{2} \)
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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

−12.05800760409234496659905379482, −10.81734159870389400283951393908, −10.11447766399339620842557054867, −9.179477767301959432965257844574, −7.63981343849217256850953612812, −6.66605832100276560221092762736, −5.46803479900841481066862649227, −4.46932696750048817302174138200, −3.45561762209284237654994176120, −1.73220669511550803654225237727, 2.22411586414352674696707961894, 3.63559836197354319758346018895, 4.70211172533508292122972380454, 6.11143791718917578801996760802, 6.85794255559593248829090854677, 7.68101455293627952322282935488, 9.017274496128808705577989534293, 10.28608257991375642785494434376, 11.25966738854932157514853096155, 12.04581091193201833336502014589

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