Properties

Degree $2$
Conductor $3024$
Sign $0.821 - 0.570i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.841 + 1.45i)5-s + (−1.65 − 2.06i)7-s + (−0.622 + 1.07i)11-s + (1.96 − 3.39i)13-s + (1.62 + 2.81i)17-s + (−2.36 + 4.09i)19-s + (0.199 + 0.344i)23-s + (1.08 − 1.87i)25-s + (3.19 + 5.54i)29-s + 0.578·31-s + (1.61 − 4.14i)35-s + (2.72 − 4.71i)37-s + (−4.20 + 7.27i)41-s + (−2.46 − 4.26i)43-s − 0.425·47-s + ⋯
L(s)  = 1  + (0.376 + 0.651i)5-s + (−0.625 − 0.780i)7-s + (−0.187 + 0.325i)11-s + (0.543 − 0.941i)13-s + (0.394 + 0.683i)17-s + (−0.541 + 0.938i)19-s + (0.0415 + 0.0718i)23-s + (0.216 − 0.375i)25-s + (0.594 + 1.02i)29-s + 0.103·31-s + (0.273 − 0.701i)35-s + (0.447 − 0.774i)37-s + (−0.656 + 1.13i)41-s + (−0.375 − 0.650i)43-s − 0.0620·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.821 - 0.570i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.821 - 0.570i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.761110949\)
\(L(\frac12)\) \(\approx\) \(1.761110949\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (1.65 + 2.06i)T \)
good5 \( 1 + (-0.841 - 1.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.622 - 1.07i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.96 + 3.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.62 - 2.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.36 - 4.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.199 - 0.344i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.19 - 5.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.578T + 31T^{2} \)
37 \( 1 + (-2.72 + 4.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.20 - 7.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.46 + 4.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.425T + 47T^{2} \)
53 \( 1 + (-0.466 - 0.807i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.05T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 - 9.41T + 67T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + (-6.82 - 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.53T + 79T^{2} \)
83 \( 1 + (8.03 + 13.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.03 + 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.86 + 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.650635573293036320314445384216, −8.063982577819911947207403597431, −7.21151890096084207975062039611, −6.50001479419567640904914378395, −5.94001873669171304345004780330, −4.98680236979219005615160673331, −3.83173388331015050974927654251, −3.30841242458757638951994098681, −2.23801183385083374500013323349, −0.957606770476427571434360764802, 0.69351962845897691422040194358, 2.03308433799188136229684796261, 2.88601056113741497118487344950, 3.92761323715287308960034876335, 4.93920004839525798124516015730, 5.50937679731944989020605522901, 6.46806836711530400555663618066, 6.88382279278991967690795095268, 8.213368988907090558903517256249, 8.628317788718120007347907323948

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