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Properties

Label 2-280-280.69-c0-0-0
Degree $2$
Conductor $280$
Sign $0.707 - 0.707i$
Analytic cond. $0.139738$
Root an. cond. $0.373815$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + 1.41i·3-s − 4-s + (−0.707 + 0.707i)5-s + 1.41·6-s + i·7-s + i·8-s − 1.00·9-s + (0.707 + 0.707i)10-s − 1.41i·12-s − 1.41i·13-s + 14-s + (−1.00 − 1.00i)15-s + 16-s + 1.00i·18-s + 1.41·19-s + ⋯
L(s)  = 1  − i·2-s + 1.41i·3-s − 4-s + (−0.707 + 0.707i)5-s + 1.41·6-s + i·7-s + i·8-s − 1.00·9-s + (0.707 + 0.707i)10-s − 1.41i·12-s − 1.41i·13-s + 14-s + (−1.00 − 1.00i)15-s + 16-s + 1.00i·18-s + 1.41·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.139738\)
Root analytic conductor: \(0.373815\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6259243436\)
\(L(\frac12)\) \(\approx\) \(0.6259243436\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 - 1.41iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.41T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.87632174711782255641358370677, −11.17899303870328303088224976438, −10.35542758146941242718469713287, −9.698222947804917371484020649155, −8.731615248253548933574620284630, −7.73410312088666485613325094090, −5.70606865385969620320556948495, −4.83123021718559643282327601472, −3.53872943663913842518785477498, −2.87554991319898098850264247325, 1.19131665193209874420607157325, 3.83008646919882056223614640972, 4.95575255671963414409236138270, 6.38754131075484860594574741791, 7.25753457813564924206943884245, 7.70262059668064359062657271905, 8.727191755292717787655958912301, 9.736460018491708411742477094034, 11.39076040648746491921282885073, 12.20898618630272905875130760286

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