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Properties

Label 2-280-280.13-c1-0-32
Degree $2$
Conductor $280$
Sign $0.904 + 0.425i$
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 − i)2-s + (1.45 + 1.45i)3-s − 2i·4-s + (2.22 − 0.254i)5-s + 2.91·6-s + (−1.87 + 1.87i)7-s + (−2 − 2i)8-s + 1.25i·9-s + (1.96 − 2.47i)10-s + (2.91 − 2.91i)12-s + (−1.45 − 1.45i)13-s + 3.74i·14-s + (3.61 + 2.87i)15-s − 4·16-s + (1.25 + 1.25i)18-s + 8.37i·19-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.842 + 0.842i)3-s − i·4-s + (0.993 − 0.113i)5-s + 1.19·6-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + 0.419i·9-s + (0.622 − 0.782i)10-s + (0.842 − 0.842i)12-s + (−0.404 − 0.404i)13-s + 0.999i·14-s + (0.932 + 0.741i)15-s − 16-s + (0.296 + 0.296i)18-s + 1.92i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.904 + 0.425i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.904 + 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31951 - 0.518293i\)
\(L(\frac12)\) \(\approx\) \(2.31951 - 0.518293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + (-2.22 + 0.254i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (-1.45 - 1.45i)T + 3iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (1.45 + 1.45i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 - 8.37iT - 19T^{2} \)
23 \( 1 + (6.74 + 6.74i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 - 11.2iT - 59T^{2} \)
61 \( 1 + 14.2T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 15.2T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 - 8.25iT - 79T^{2} \)
83 \( 1 + (4.00 + 4.00i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−12.18380704978376925843266785923, −10.44074589644682536391537566699, −10.02232502492269369970380748303, −9.328400894705008068761024978942, −8.362369895419034647309157380014, −6.33799779449818207557698234653, −5.63384696892290555226808251671, −4.30285224622664934297551790744, −3.16868476409146693255387797604, −2.16704041827730381021881526940, 2.19328723600504689848836826335, 3.32265975946474870694694836611, 4.85483556443704950444838208912, 6.20160252021591302388346558828, 6.99980899977837401968250039117, 7.68955549738150331787418957384, 8.943610733493518492089135160513, 9.721467897962077138392045503393, 11.15231042679865955968164792671, 12.44979333669003403388106713284

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