↑

Properties

Label 2-1856-116.111-c0-0-0
Degree $2$
Conductor $1856$
Sign $0.353 - 0.935i$
Analytic cond. $0.926264$
Root an. cond. $0.962426$
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  + (1.12 + 1.40i)5-s + (−0.900 + 0.433i)9-s + (−0.400 − 0.193i)13-s + 1.24·17-s + (−0.500 + 2.19i)25-s + (0.900 + 0.433i)29-s + (−0.400 + 0.193i)37-s − 0.445·41-s + (−1.62 − 0.781i)45-s + (−0.900 + 0.433i)49-s + (−0.777 − 0.974i)53-s + (0.277 + 1.21i)61-s + (−0.178 − 0.781i)65-s + (0.777 − 0.974i)73-s + (0.623 − 0.781i)81-s + ⋯
L(s)  = 1  + (1.12 + 1.40i)5-s + (−0.900 + 0.433i)9-s + (−0.400 − 0.193i)13-s + 1.24·17-s + (−0.500 + 2.19i)25-s + (0.900 + 0.433i)29-s + (−0.400 + 0.193i)37-s − 0.445·41-s + (−1.62 − 0.781i)45-s + (−0.900 + 0.433i)49-s + (−0.777 − 0.974i)53-s + (0.277 + 1.21i)61-s + (−0.178 − 0.781i)65-s + (0.777 − 0.974i)73-s + (0.623 − 0.781i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $0.353 - 0.935i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ 0.353 - 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.288888866\)
\(L(\frac12)\) \(\approx\) \(1.288888866\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-0.900 - 0.433i)T \)
good3 \( 1 + (0.900 - 0.433i)T^{2} \)
5 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (0.400 + 0.193i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 - 1.24T + T^{2} \)
19 \( 1 + (0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.222 - 0.974i)T^{2} \)
37 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
41 \( 1 + 0.445T + T^{2} \)
43 \( 1 + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 + (-0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
79 \( 1 + (-0.623 + 0.781i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
97 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
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

−9.887924595573951873047106980895, −8.844676183254137268957967537114, −7.932690538788205482651195851311, −7.14869728700933275009506880421, −6.33710375767659731649440012204, −5.69004735769575197629029936385, −4.94719996369427809411741744011, −3.31217466604819727718733129769, −2.83709722240483105810157874761, −1.79133123648937081140319502053, 1.00529995911701544077670993493, 2.16319791149616451417262272536, 3.30840804744627297131442995630, 4.58032189240322875801189173787, 5.32300274845902833745863237052, 5.88380350390551735147761391206, 6.72420189309195261509417827580, 8.053469243132613273779483288832, 8.485866928906442214636313954699, 9.473003832949264127401230254894

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