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Properties

Label 2-280-280.107-c1-0-41
Degree $2$
Conductor $280$
Sign $-0.754 + 0.656i$
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  + (1.41 − 0.0934i)2-s + (−2.34 − 0.629i)3-s + (1.98 − 0.263i)4-s + (−2.03 − 0.927i)5-s + (−3.37 − 0.668i)6-s + (−2.59 − 0.511i)7-s + (2.77 − 0.557i)8-s + (2.52 + 1.45i)9-s + (−2.95 − 1.11i)10-s + (−2.68 − 4.65i)11-s + (−4.82 − 0.628i)12-s + (−0.646 − 0.646i)13-s + (−3.71 − 0.479i)14-s + (4.19 + 3.45i)15-s + (3.86 − 1.04i)16-s + (−0.0172 + 0.0643i)17-s + ⋯
L(s)  = 1  + (0.997 − 0.0660i)2-s + (−1.35 − 0.363i)3-s + (0.991 − 0.131i)4-s + (−0.909 − 0.414i)5-s + (−1.37 − 0.273i)6-s + (−0.981 − 0.193i)7-s + (0.980 − 0.197i)8-s + (0.841 + 0.485i)9-s + (−0.935 − 0.353i)10-s + (−0.810 − 1.40i)11-s + (−1.39 − 0.181i)12-s + (−0.179 − 0.179i)13-s + (−0.991 − 0.128i)14-s + (1.08 + 0.893i)15-s + (0.965 − 0.261i)16-s + (−0.00418 + 0.0156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.754 + 0.656i$
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: \((2,\ 280,\ (\ :1/2),\ -0.754 + 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.285555 - 0.762714i\)
\(L(\frac12)\) \(\approx\) \(0.285555 - 0.762714i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0934i)T \)
5 \( 1 + (2.03 + 0.927i)T \)
7 \( 1 + (2.59 + 0.511i)T \)
good3 \( 1 + (2.34 + 0.629i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.68 + 4.65i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.646 + 0.646i)T + 13iT^{2} \)
17 \( 1 + (0.0172 - 0.0643i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.477 + 0.275i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.32 - 1.42i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.78T + 29T^{2} \)
31 \( 1 + (-8.21 + 4.74i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.53 + 5.71i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.40T + 41T^{2} \)
43 \( 1 + (2.49 - 2.49i)T - 43iT^{2} \)
47 \( 1 + (2.23 + 8.32i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.213 + 0.798i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.94 - 3.43i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.60 - 2.08i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.87 - 7.00i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (3.97 + 1.06i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.94 - 13.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.13 + 9.13i)T - 83iT^{2} \)
89 \( 1 + (12.2 + 7.05i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.04 + 6.04i)T + 97iT^{2} \)
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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

−11.69873564558817467816615680649, −10.96700018358404257453524728680, −10.11750954287335671223850750548, −8.283935878825237763079007490081, −7.22905785851064141481421646612, −6.16382851518990844722654374983, −5.55904730933185118852207285782, −4.32570955545511625011891311298, −3.08985154240989898332032423030, −0.50527847605590030324551382318, 2.77805798951401601217064690378, 4.24803437216421129117662108965, 4.97393504336638087729195036193, 6.24767186087581632915103485240, 6.86159657641883726894865767668, 7.974321572796766097650456896841, 10.03925772955863218893745251729, 10.45965420186941669678764832691, 11.57496490593393326762226744317, 12.25193458306597542532350144467

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