↑

Properties

Label 2-2912-364.51-c0-0-0
Degree $2$
Conductor $2912$
Sign $0.197 - 0.980i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
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)3-s + (−0.866 − 0.5i)5-s − 7-s + (−0.5 − 0.866i)11-s + 13-s + 0.999·15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)21-s + (−0.866 − 0.5i)23-s − i·27-s + (0.5 + 0.866i)31-s + (0.866 + 0.499i)33-s + (0.866 + 0.5i)35-s + (−0.866 − 0.5i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.866 − 0.5i)5-s − 7-s + (−0.5 − 0.866i)11-s + 13-s + 0.999·15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)21-s + (−0.866 − 0.5i)23-s − i·27-s + (0.5 + 0.866i)31-s + (0.866 + 0.499i)33-s + (0.866 + 0.5i)35-s + (−0.866 − 0.5i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.197 - 0.980i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :0),\ 0.197 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4589619486\)
\(L(\frac12)\) \(\approx\) \(0.4589619486\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
13 \( 1 - T \)
good3 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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

−9.074194690380133340639334076787, −8.224885753134341190203907937793, −7.894617960890736863506208435254, −6.46728663337270782006890983262, −6.14412316573716145055105993094, −5.24779949924084907972796422112, −4.45806466071811924043723833379, −3.63824258602976444574553503370, −2.85269796237646499722716536620, −0.953532756705708350182267443491, 0.41973649853521409597449471018, 1.99767277119106943672989819978, 3.47333933463075316706528962319, 3.67707373399344166668005155508, 5.13897002112771356812991167224, 5.81803174434588399968681936424, 6.54729330112640457409641588069, 7.26258637062818815694778718031, 7.69581834463195570291595552568, 8.742181666296059642912779016340

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