↑

Properties

Label 2-1560-1560.1403-c0-0-15
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 + 0.382i)2-s + (0.707 + 0.707i)3-s + (0.707 + 0.707i)4-s + (0.382 − 0.923i)5-s + (0.382 + 0.923i)6-s + (0.382 + 0.923i)8-s + 1.00i·9-s + (0.707 − 0.707i)10-s − 1.84·11-s + i·12-s + (0.707 − 0.707i)13-s + (0.923 − 0.382i)15-s + i·16-s + (−0.382 + 0.923i)18-s + (0.923 − 0.382i)20-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)2-s + (0.707 + 0.707i)3-s + (0.707 + 0.707i)4-s + (0.382 − 0.923i)5-s + (0.382 + 0.923i)6-s + (0.382 + 0.923i)8-s + 1.00i·9-s + (0.707 − 0.707i)10-s − 1.84·11-s + i·12-s + (0.707 − 0.707i)13-s + (0.923 − 0.382i)15-s + i·16-s + (−0.382 + 0.923i)18-s + (0.923 − 0.382i)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: $\chi_{1560} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.373876572\)
\(L(\frac12)\) \(\approx\) \(2.373876572\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 - 0.382i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + 1.84T + T^{2} \)
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 \( 1 - 0.765T + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 0.765iT - T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.84iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
89 \( 1 - 1.84iT - T^{2} \)
97 \( 1 + iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.763673799756293318120286833419, −8.730911112421700855270147783204, −8.084809603351847383504296666306, −7.65155648192063212717038828520, −6.23826340579759278980009614942, −5.21460546762718955963022131618, −5.05698651229284038333836895809, −3.90365071498691143984784925395, −2.99144432707594558910151211399, −2.06177190652976047698193965960, 1.67722487981037798104266740889, 2.62207775017757905383560083936, 3.17222457461606376476687522728, 4.28673304593837246257309732499, 5.52396096755768833074596797650, 6.20965814724593940114544523626, 7.03470007496358397332716907787, 7.65325404359676388504374366014, 8.622798402181741786133177316647, 9.765833886599392280571018462028

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