↑

Properties

Label 2-992-992.805-c0-0-2
Degree $2$
Conductor $992$
Sign $-0.896 + 0.442i$
Analytic cond. $0.495072$
Root an. cond. $0.703613$
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 + (0.499 − 0.866i)4-s + (−0.241 − 0.0999i)5-s + (−1.22 − 1.22i)7-s + 0.999i·8-s + (−0.707 + 0.707i)9-s + (0.258 − 0.0340i)10-s + (1.67 + 0.448i)14-s + (−0.5 − 0.866i)16-s + (0.258 − 0.965i)18-s + (−1.83 + 0.758i)19-s + (−0.207 + 0.158i)20-s + (−0.658 − 0.658i)25-s + (−1.67 + 0.448i)28-s − 31-s + (0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.241 − 0.0999i)5-s + (−1.22 − 1.22i)7-s + 0.999i·8-s + (−0.707 + 0.707i)9-s + (0.258 − 0.0340i)10-s + (1.67 + 0.448i)14-s + (−0.5 − 0.866i)16-s + (0.258 − 0.965i)18-s + (−1.83 + 0.758i)19-s + (−0.207 + 0.158i)20-s + (−0.658 − 0.658i)25-s + (−1.67 + 0.448i)28-s − 31-s + (0.866 + 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(992\)    =    \(2^{5} \cdot 31\)
Sign: $-0.896 + 0.442i$
Analytic conductor: \(0.495072\)
Root analytic conductor: \(0.703613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{992} (805, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 992,\ (\ :0),\ -0.896 + 0.442i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07167084987\)
\(L(\frac12)\) \(\approx\) \(0.07167084987\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + T \)
good3 \( 1 + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.241 + 0.0999i)T + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (1.83 - 0.758i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-1.46 - 0.607i)T + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + 0.517T + T^{2} \)
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

−9.990646241565693467814282256977, −8.867946915040615560680658915927, −8.180934080647338272225112459830, −7.39052753592464010457682501633, −6.53667891723286909503362883655, −5.89459472712328843950952749979, −4.59495207741106579565552232645, −3.45569465573556072704051124869, −2.02709464748021374910338050834, −0.081120722375025056706134284974, 2.20274686462844145518511045151, 3.06035654267805388762299321783, 3.94389679363142864538751909419, 5.66611928212907116145597883048, 6.43567830761829499523445393987, 7.18341541329133039058910844354, 8.552704186888560069413252107115, 8.852329826012100062980072453138, 9.571203957891309013805339935799, 10.42884705835808184915490326462

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