↑

Properties

Label 2-1476-123.32-c0-0-0
Degree $2$
Conductor $1476$
Sign $0.485 - 0.874i$
Analytic cond. $0.736619$
Root an. cond. $0.858265$
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  − 1.41·5-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)17-s + (1 + i)19-s + 1.00·25-s + (0.707 + 0.707i)29-s − 31-s + 37-s + (0.707 + 0.707i)41-s + i·43-s + (0.707 − 0.707i)47-s + i·49-s + (−1.00 − 1.00i)55-s + i·61-s + (−1 − i)67-s + ⋯
L(s)  = 1  − 1.41·5-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)17-s + (1 + i)19-s + 1.00·25-s + (0.707 + 0.707i)29-s − 31-s + 37-s + (0.707 + 0.707i)41-s + i·43-s + (0.707 − 0.707i)47-s + i·49-s + (−1.00 − 1.00i)55-s + i·61-s + (−1 − i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1476\)    =    \(2^{2} \cdot 3^{2} \cdot 41\)
Sign: $0.485 - 0.874i$
Analytic conductor: \(0.736619\)
Root analytic conductor: \(0.858265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1476} (1385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1476,\ (\ :0),\ 0.485 - 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8004816110\)
\(L(\frac12)\) \(\approx\) \(0.8004816110\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 + (-0.707 - 0.707i)T \)
good5 \( 1 + 1.41T + T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T + T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - iT - T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + (-1 + i)T - iT^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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

−9.743760571943040412088507712514, −8.998616529623702813085279124484, −8.086952237974868603408260765916, −7.52962445782369230526241590507, −6.74923924152430453369298963805, −5.76069325817840408685751921537, −4.49882591202135385695074938502, −4.01180857281692616062435471434, −3.01552189197917663611260438185, −1.44206010273922956149895223690, 0.72336711461650633457673066807, 2.62340723348072100197295769917, 3.63999250369576898041391731541, 4.34282223332171369858525347135, 5.31785870282559752922319429936, 6.45938969197599088029296484244, 7.27504991580990713276829128959, 7.84853294510522158072603224088, 8.865650182033123224426772914001, 9.264026730253667020220740888018

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