↑

Properties

Label 2-1140-1140.1139-c0-0-17
Degree $2$
Conductor $1140$
Sign $0.382 + 0.923i$
Analytic cond. $0.568934$
Root an. cond. $0.754277$
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 + (0.382 − 0.923i)3-s + (0.707 − 0.707i)4-s + i·5-s − i·6-s + (0.382 − 0.923i)8-s + (−0.707 − 0.707i)9-s + (0.382 + 0.923i)10-s + 1.41·11-s + (−0.382 − 0.923i)12-s − 1.84·13-s + (0.923 + 0.382i)15-s − i·16-s + (−0.923 − 0.382i)18-s + i·19-s + (0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (0.382 − 0.923i)3-s + (0.707 − 0.707i)4-s + i·5-s − i·6-s + (0.382 − 0.923i)8-s + (−0.707 − 0.707i)9-s + (0.382 + 0.923i)10-s + 1.41·11-s + (−0.382 − 0.923i)12-s − 1.84·13-s + (0.923 + 0.382i)15-s − i·16-s + (−0.923 − 0.382i)18-s + i·19-s + (0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(0.568934\)
Root analytic conductor: \(0.754277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.962586581\)
\(L(\frac12)\) \(\approx\) \(1.962586581\)
\(L(1)\) 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 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 - iT \)
19 \( 1 - iT \)
good7 \( 1 + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 + 1.84T + T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 0.765T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.84iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + 0.765iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 0.765T + T^{2} \)
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

−9.876382292510239357472500689841, −9.297096675762447656445541502611, −7.80111183840593379739173315361, −7.21967984331049100173319494727, −6.47090536277124673995840813467, −5.84573042582163858766664249386, −4.47949506418939143892777996852, −3.44758680955968959380050527008, −2.60652447065316114440653357938, −1.64203278993035990362370137013, 2.11885722183291503816858181558, 3.28347426441487786417926405365, 4.40632784498783062530720833810, 4.73912603855112715802295414016, 5.63305024373900327211904741301, 6.75712584952729113886295691988, 7.67763354442140431718586240038, 8.568561496632540432789848131104, 9.332956671895512253503071317015, 9.921909417600087357339625228514

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