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Properties

Label 2-1560-1560.389-c0-0-5
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

\(L(\frac{1}{2})\) \(\approx\) \(0.6371074505\)
\(L(\frac12)\) \(\approx\) \(0.6371074505\)
\(L(1)\) 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

−9.280415348874297448936626358500, −8.570554956129462779228584161184, −8.047314160610133101530080005359, −6.49147006486127271139307636863, −5.65791256348840299180803701653, −5.14637900411488151018799580618, −4.05881082455783942215491415621, −3.33109429810329716051787720410, −1.60489032569598657217205765218, −0.62866525652835745724448548805, 1.72960181570808695176212245361, 3.51548440099056782291667955892, 4.47929764363723737965942924075, 5.17028030406533126560967941397, 6.31658448585394989182106715003, 6.61618216905513338139999218469, 7.41310843047231011074407866626, 8.076897121303921053203726152889, 9.465881515785681660784069950311, 9.891767780930705834507708458391

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