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Properties

Label 2-280-280.107-c1-0-26
Degree $2$
Conductor $280$
Sign $-0.978 + 0.208i$
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 − 0.0262i)2-s + (−2.91 − 0.782i)3-s + (1.99 + 0.0742i)4-s + (0.0325 − 2.23i)5-s + (4.10 + 1.18i)6-s + (2.00 − 1.72i)7-s + (−2.82 − 0.157i)8-s + (5.31 + 3.06i)9-s + (−0.104 + 3.16i)10-s + (−0.678 − 1.17i)11-s + (−5.77 − 1.78i)12-s + (0.979 + 0.979i)13-s + (−2.88 + 2.38i)14-s + (−1.84 + 6.50i)15-s + (3.98 + 0.296i)16-s + (1.32 − 4.95i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0185i)2-s + (−1.68 − 0.451i)3-s + (0.999 + 0.0371i)4-s + (0.0145 − 0.999i)5-s + (1.67 + 0.482i)6-s + (0.757 − 0.652i)7-s + (−0.998 − 0.0556i)8-s + (1.77 + 1.02i)9-s + (−0.0331 + 0.999i)10-s + (−0.204 − 0.354i)11-s + (−1.66 − 0.513i)12-s + (0.271 + 0.271i)13-s + (−0.769 + 0.638i)14-s + (−0.476 + 1.67i)15-s + (0.997 + 0.0742i)16-s + (0.321 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.978 + 0.208i$
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: \((2,\ 280,\ (\ :1/2),\ -0.978 + 0.208i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0334376 - 0.317150i\)
\(L(\frac12)\) \(\approx\) \(0.0334376 - 0.317150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0262i)T \)
5 \( 1 + (-0.0325 + 2.23i)T \)
7 \( 1 + (-2.00 + 1.72i)T \)
good3 \( 1 + (2.91 + 0.782i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (0.678 + 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.979 - 0.979i)T + 13iT^{2} \)
17 \( 1 + (-1.32 + 4.95i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.73 + 2.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.35 - 2.23i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 0.988T + 29T^{2} \)
31 \( 1 + (-0.611 + 0.352i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.58 - 9.65i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.56T + 41T^{2} \)
43 \( 1 + (-2.14 + 2.14i)T - 43iT^{2} \)
47 \( 1 + (-0.893 - 3.33i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.16 + 4.33i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.09 + 2.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.03 + 5.21i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0191 + 0.0713i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.469iT - 71T^{2} \)
73 \( 1 + (7.49 + 2.00i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.32 - 2.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.79 - 1.79i)T - 83iT^{2} \)
89 \( 1 + (7.18 + 4.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.40 + 1.40i)T + 97iT^{2} \)
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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

−11.54685997279955096253249919677, −10.60005901905532000746274401814, −9.720612700928796183644317131705, −8.389342597875140703961481847725, −7.55296151417743677628907844868, −6.50102080345665914579608632573, −5.53057551275963814503215838372, −4.47167499811841584892664109484, −1.64276220048313049007380194397, −0.40993470509111299979254143812, 1.98379305814568531969006244238, 4.09284154443588273274941098040, 5.90048975443202769636008537029, 6.04316464816051080970824250439, 7.41047581073026581753122943786, 8.437394660246078213933804913570, 9.935482263742488648843100037859, 10.53562723772272026021692679547, 11.03391686240038996469092792383, 11.97048266875041658618861760037

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