↑

Properties

Label 2-1155-1155.524-c0-0-0
Degree $2$
Conductor $1155$
Sign $-0.997 - 0.0694i$
Analytic cond. $0.576420$
Root an. cond. $0.759223$
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  + i·3-s + (0.309 + 0.951i)4-s + (−0.587 − 0.809i)5-s + (−0.951 + 0.309i)7-s − 9-s + (−0.809 − 0.587i)11-s + (−0.951 + 0.309i)12-s + (−0.363 + 0.5i)13-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (−0.951 + 0.690i)17-s + (0.587 − 0.809i)20-s + (−0.309 − 0.951i)21-s + (−0.309 + 0.951i)25-s − i·27-s + (−0.587 − 0.809i)28-s + ⋯
L(s)  = 1  + i·3-s + (0.309 + 0.951i)4-s + (−0.587 − 0.809i)5-s + (−0.951 + 0.309i)7-s − 9-s + (−0.809 − 0.587i)11-s + (−0.951 + 0.309i)12-s + (−0.363 + 0.5i)13-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (−0.951 + 0.690i)17-s + (0.587 − 0.809i)20-s + (−0.309 − 0.951i)21-s + (−0.309 + 0.951i)25-s − i·27-s + (−0.587 − 0.809i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.997 - 0.0694i$
Analytic conductor: \(0.576420\)
Root analytic conductor: \(0.759223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :0),\ -0.997 - 0.0694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4776811720\)
\(L(\frac12)\) \(\approx\) \(0.4776811720\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
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

−10.49539366312621119022810451944, −9.314836679927468190802505205619, −8.798557681115265291250086531036, −8.215275462825029930711633614337, −7.17354950687535829504951479928, −6.15265116588465976147546472217, −5.10507154404155927401686225019, −4.19903367632250260719653735966, −3.43936272438415177248638078228, −2.53631638899935148784645484442, 0.38632136365402133628361192920, 2.31126491365550684933553375934, 2.87864373818851660644446417177, 4.40779551606272287050418061375, 5.64677913740131643840503835283, 6.38713241507640976284667059486, 7.15937846825700682522371724804, 7.52065833154657709685021784246, 8.717918402615992931806678476604, 9.867792746312689643254759790090

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