↑

Properties

Label 2-1260-1260.839-c0-0-3
Degree $2$
Conductor $1260$
Sign $0.939 - 0.342i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
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  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·6-s + i·7-s + 0.999i·8-s + (0.499 − 0.866i)9-s − 0.999i·10-s + (0.866 + 0.499i)12-s + (−0.5 + 0.866i)14-s + (−0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.499i)18-s + (0.499 − 0.866i)20-s + (0.5 + 0.866i)21-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·6-s + i·7-s + 0.999i·8-s + (0.499 − 0.866i)9-s − 0.999i·10-s + (0.866 + 0.499i)12-s + (−0.5 + 0.866i)14-s + (−0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.499i)18-s + (0.499 − 0.866i)20-s + (0.5 + 0.866i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.095615475\)
\(L(\frac12)\) \(\approx\) \(2.095615475\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 - iT \)
good11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−9.442413488919182438128448232098, −8.925929984984069099324445688062, −8.144646515472830837606652272419, −7.60549154046496127531641376828, −6.61876952284989624106542717003, −5.71288777379053705001596430561, −4.82196709282594845077544206950, −3.87151233419364177439080981761, −2.92794546034471601862078544240, −1.85297912758672435225675914058, 1.76400738882299278238276240963, 3.08842417650883422307996263264, 3.56956795292172264680932431740, 4.39777872334338937201172873753, 5.32667596367070540388568609379, 6.70819433214857338657820417591, 7.25051909023950680412538339688, 8.077603836224562347064773778918, 9.345205719252276185738233217486, 9.978860845195758042259936251089

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