Properties

Label 2-24e2-16.13-c1-0-0
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

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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: $\chi_{576} (145, \cdot )$
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 \( 1 + 0.571T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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