Properties

Label 2-280-280.107-c1-0-37
Degree $2$
Conductor $280$
Sign $0.865 + 0.501i$
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  + (1.37 − 0.309i)2-s + (0.821 + 0.220i)3-s + (1.80 − 0.854i)4-s + (−0.801 − 2.08i)5-s + (1.20 + 0.0493i)6-s + (1.14 + 2.38i)7-s + (2.23 − 1.73i)8-s + (−1.97 − 1.13i)9-s + (−1.75 − 2.63i)10-s + (0.933 + 1.61i)11-s + (1.67 − 0.303i)12-s + (−1.15 − 1.15i)13-s + (2.31 + 2.93i)14-s + (−0.198 − 1.89i)15-s + (2.53 − 3.09i)16-s + (−1.40 + 5.24i)17-s + ⋯
L(s)  = 1  + (0.975 − 0.218i)2-s + (0.474 + 0.127i)3-s + (0.904 − 0.427i)4-s + (−0.358 − 0.933i)5-s + (0.490 + 0.0201i)6-s + (0.432 + 0.901i)7-s + (0.788 − 0.614i)8-s + (−0.657 − 0.379i)9-s + (−0.553 − 0.832i)10-s + (0.281 + 0.487i)11-s + (0.482 − 0.0877i)12-s + (−0.321 − 0.321i)13-s + (0.619 + 0.784i)14-s + (−0.0512 − 0.488i)15-s + (0.634 − 0.772i)16-s + (−0.340 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.865 + 0.501i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.865 + 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34477 - 0.630090i\)
\(L(\frac12)\) \(\approx\) \(2.34477 - 0.630090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.309i)T \)
5 \( 1 + (0.801 + 2.08i)T \)
7 \( 1 + (-1.14 - 2.38i)T \)
good3 \( 1 + (-0.821 - 0.220i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.933 - 1.61i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.15 + 1.15i)T + 13iT^{2} \)
17 \( 1 + (1.40 - 5.24i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.58 + 0.915i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.26 + 0.605i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.08T + 29T^{2} \)
31 \( 1 + (5.06 - 2.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.55 - 9.54i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.49T + 41T^{2} \)
43 \( 1 + (1.18 - 1.18i)T - 43iT^{2} \)
47 \( 1 + (1.20 + 4.50i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.06 + 11.4i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-7.77 + 4.48i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.16 + 4.71i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.871 + 3.25i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.09iT - 71T^{2} \)
73 \( 1 + (8.12 + 2.17i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.84 - 10.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.95 + 4.95i)T - 83iT^{2} \)
89 \( 1 + (-7.31 - 4.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.17 - 6.17i)T + 97iT^{2} \)
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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

−11.98612959942090625850268980351, −11.22391027616429258349410285767, −9.913937519191640477642527902578, −8.770537208875468867820673616654, −8.117258919263289365632649899305, −6.58304954905047105563690958780, −5.44667140920240991653695735390, −4.56547346062970652121995019972, −3.33546375965161289135942151697, −1.90263094838562537196934251458, 2.40378618075371598271773551167, 3.48284970333419029359788788795, 4.59413981831390907440932294353, 5.94995481315886163055896607818, 7.19808888841949925269049069480, 7.58999247701212373732030332673, 8.859676782846005419668799655850, 10.46388870344734018822636483859, 11.22281492264054836813037739286, 11.77560836578151588003242624097

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