↑

Properties

Label 2-24e2-16.5-c1-0-8
Degree $2$
Conductor $576$
Sign $-0.969 + 0.243i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.27 − 1.27i)5-s + 0.158i·7-s + (−3.79 − 3.79i)11-s + (−4.21 + 4.21i)13-s − 3.05·17-s + (2.15 − 2.15i)19-s + 2.82i·23-s − 1.76i·25-s + (−2.09 + 2.09i)29-s − 4.15·31-s + (0.202 − 0.202i)35-s + (−5.98 − 5.98i)37-s + 2.60i·41-s + (−5.75 − 5.75i)43-s − 2.82·47-s + ⋯
L(s)  = 1  + (−0.568 − 0.568i)5-s + 0.0600i·7-s + (−1.14 − 1.14i)11-s + (−1.16 + 1.16i)13-s − 0.740·17-s + (0.495 − 0.495i)19-s + 0.589i·23-s − 0.353i·25-s + (−0.389 + 0.389i)29-s − 0.746·31-s + (0.0341 − 0.0341i)35-s + (−0.984 − 0.984i)37-s + 0.406i·41-s + (−0.877 − 0.877i)43-s − 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.969 + 0.243i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.969 + 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0381619 - 0.309043i\)
\(L(\frac12)\) \(\approx\) \(0.0381619 - 0.309043i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1.27 + 1.27i)T + 5iT^{2} \)
7 \( 1 - 0.158iT - 7T^{2} \)
11 \( 1 + (3.79 + 3.79i)T + 11iT^{2} \)
13 \( 1 + (4.21 - 4.21i)T - 13iT^{2} \)
17 \( 1 + 3.05T + 17T^{2} \)
19 \( 1 + (-2.15 + 2.15i)T - 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (2.09 - 2.09i)T - 29iT^{2} \)
31 \( 1 + 4.15T + 31T^{2} \)
37 \( 1 + (5.98 + 5.98i)T + 37iT^{2} \)
41 \( 1 - 2.60iT - 41T^{2} \)
43 \( 1 + (5.75 + 5.75i)T + 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (3.55 + 3.55i)T + 53iT^{2} \)
59 \( 1 + (-4 - 4i)T + 59iT^{2} \)
61 \( 1 + (-3.66 + 3.66i)T - 61iT^{2} \)
67 \( 1 + (0.767 - 0.767i)T - 67iT^{2} \)
71 \( 1 - 0.317iT - 71T^{2} \)
73 \( 1 + 1.33iT - 73T^{2} \)
79 \( 1 - 9.69T + 79T^{2} \)
83 \( 1 + (-0.115 + 0.115i)T - 83iT^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + 0.571T + 97T^{2} \)
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

−10.39189510372635667303642800820, −9.263397782806361845221860843563, −8.635325404495458687939489311552, −7.66155666922803398947172862471, −6.86071866532781198398779888253, −5.49048350341989699844066978107, −4.76377428370256359804378217332, −3.56541731949099158182023770368, −2.21292312351046014881160479739, −0.16229687653780051363105899469, 2.26716528070121439413480142687, 3.30873295522447440009035064494, 4.67055929854262546106178777833, 5.46070311411172374213420953693, 6.91924989167466146188659499027, 7.52780000004701945272783859456, 8.234316708533570855054481706506, 9.625454084020087114240134376681, 10.27246063758680064413298156402, 10.97986215608111861693260416189

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