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Properties

Label 2-1560-1560.389-c0-0-1
Degree $2$
Conductor $1560$
Sign $1$
Analytic cond. $0.778541$
Root an. cond. $0.882349$
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 + 3-s − 4-s + i·5-s − i·6-s + i·8-s + 9-s + 10-s + 2i·11-s − 12-s − 13-s + i·15-s + 16-s − i·18-s − i·20-s + ⋯
L(s)  = 1  − i·2-s + 3-s − 4-s + i·5-s − i·6-s + i·8-s + 9-s + 10-s + 2i·11-s − 12-s − 13-s + i·15-s + 16-s − i·18-s − i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.778541\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.339258904\)
\(L(\frac12)\) \(\approx\) \(1.339258904\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - T \)
5 \( 1 - iT \)
13 \( 1 + T \)
good7 \( 1 + T^{2} \)
11 \( 1 - 2iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 + T^{2} \)
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

−9.753678894947544096918077391793, −9.215331788080733192538079810061, −8.010789460222424734526213828205, −7.42310077696491546057527438263, −6.69522627309147958766583051317, −5.11044214664557342335684491487, −4.32159033834958989921347925136, −3.46562526957456467852610191516, −2.39713758804699978863855600071, −1.97778442303459940778389287010, 1.02393237762525616901990530172, 2.80648832756985808298729551360, 3.85562289698351112360278099844, 4.66900578016880952561193356633, 5.59113483331459689933146340066, 6.36276545415247525501692626946, 7.60924929754129671767513472710, 7.951855197918819861909315293497, 8.880019442663406608665505714210, 9.125593981196085495386480339925

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