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Properties

Label 2-1560-1560.467-c0-0-9
Degree $2$
Conductor $1560$
Sign $0.525 + 0.850i$
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  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)3-s + (−0.707 + 0.707i)4-s + (−0.923 + 0.382i)5-s + (−0.923 − 0.382i)6-s + (−0.923 − 0.382i)8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s − 0.765·11-s − i·12-s + (−0.707 − 0.707i)13-s + (0.382 − 0.923i)15-s − i·16-s + (0.923 − 0.382i)18-s + (0.382 − 0.923i)20-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)3-s + (−0.707 + 0.707i)4-s + (−0.923 + 0.382i)5-s + (−0.923 − 0.382i)6-s + (−0.923 − 0.382i)8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s − 0.765·11-s − i·12-s + (−0.707 − 0.707i)13-s + (0.382 − 0.923i)15-s − i·16-s + (0.923 − 0.382i)18-s + (0.382 − 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(0.778541\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03600595267\)
\(L(\frac12)\) \(\approx\) \(0.03600595267\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 0.923i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + 0.765T + T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 1.84T + T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + 1.84iT - T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 0.765iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
89 \( 1 + 0.765iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.599448121608625535640581947664, −8.366580808764143767511821032633, −7.913182057958590935668531654454, −6.93042762632931199317700738666, −6.31553138430115086075046112170, −5.10103490551865509503419756936, −4.87542355465153552716459209181, −3.67899003066210673536977811417, −3.01605514531165697726192175769, −0.02752846837848939101940541531, 1.53247708731844805900247322929, 2.66010211980967207248313129505, 3.85871012120193172970736729992, 4.88750740784330323689534989067, 5.27821746417602055094332674090, 6.50074893205652887628610037362, 7.33053309586402376528658827347, 8.210241460526578833838617748031, 8.950516818072257689221149466374, 10.09937457137542853964339370872

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