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Properties

Label 2-280-5.4-c5-0-13
Degree $2$
Conductor $280$
Sign $-0.469 - 0.883i$
Analytic cond. $44.9074$
Root an. cond. $6.70130$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 26.0i·3-s + (−26.2 − 49.3i)5-s + 49i·7-s − 436.·9-s + 711.·11-s − 1.00e3i·13-s + (1.28e3 − 683. i)15-s + 1.70e3i·17-s + 1.91e3·19-s − 1.27e3·21-s + 3.81e3i·23-s + (−1.74e3 + 2.58e3i)25-s − 5.05e3i·27-s + 3.52e3·29-s − 4.04e3·31-s + ⋯
L(s)  = 1  + 1.67i·3-s + (−0.469 − 0.883i)5-s + 0.377i·7-s − 1.79·9-s + 1.77·11-s − 1.65i·13-s + (1.47 − 0.784i)15-s + 1.43i·17-s + 1.21·19-s − 0.632·21-s + 1.50i·23-s + (−0.559 + 0.828i)25-s − 1.33i·27-s + 0.777·29-s − 0.755·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.469 - 0.883i$
Analytic conductor: \(44.9074\)
Root analytic conductor: \(6.70130\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :5/2),\ -0.469 - 0.883i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.840341539\)
\(L(\frac12)\) \(\approx\) \(1.840341539\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (26.2 + 49.3i)T \)
7 \( 1 - 49iT \)
good3 \( 1 - 26.0iT - 243T^{2} \)
11 \( 1 - 711.T + 1.61e5T^{2} \)
13 \( 1 + 1.00e3iT - 3.71e5T^{2} \)
17 \( 1 - 1.70e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.91e3T + 2.47e6T^{2} \)
23 \( 1 - 3.81e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.52e3T + 2.05e7T^{2} \)
31 \( 1 + 4.04e3T + 2.86e7T^{2} \)
37 \( 1 + 6.34e3iT - 6.93e7T^{2} \)
41 \( 1 - 3.36e3T + 1.15e8T^{2} \)
43 \( 1 - 7.74iT - 1.47e8T^{2} \)
47 \( 1 - 7.47e3iT - 2.29e8T^{2} \)
53 \( 1 + 7.61e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.07e4T + 7.14e8T^{2} \)
61 \( 1 + 3.42e4T + 8.44e8T^{2} \)
67 \( 1 - 4.23e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.10e4T + 1.80e9T^{2} \)
73 \( 1 + 5.38e3iT - 2.07e9T^{2} \)
79 \( 1 + 5.47e3T + 3.07e9T^{2} \)
83 \( 1 - 3.12e4iT - 3.93e9T^{2} \)
89 \( 1 - 2.53e4T + 5.58e9T^{2} \)
97 \( 1 - 6.00e4iT - 8.58e9T^{2} \)
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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

−11.28659188466008338906042524158, −10.23874460946757221235035571001, −9.369754120071252802182616110300, −8.796147755840132576765488339071, −7.75862417096132051732118941032, −5.87217742702247560609457062561, −5.17952774377948260449694506914, −3.95514592415180164011899544399, −3.42138392079361631356403638299, −1.13975789508088119970262117512, 0.60287525681054035563628802796, 1.72071687333095428338303404125, 2.99206825645812053212956633981, 4.36731053144520171152802134966, 6.29969205247079320135844072683, 6.90575843564027903390845581692, 7.31958788421109641159675679113, 8.621343086603898869133834729070, 9.599879082676025983170495023519, 11.18759599353112629689627682442

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