Properties

Label 2-24e2-16.5-c1-0-7
Degree $2$
Conductor $576$
Sign $-0.154 + 0.987i$
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  + (0.334 + 0.334i)5-s − 4.55i·7-s + (−2.47 − 2.47i)11-s + (−0.0594 + 0.0594i)13-s − 3.61·17-s + (−2.55 + 2.55i)19-s − 2.82i·23-s − 4.77i·25-s + (5.16 − 5.16i)29-s + 0.557·31-s + (1.52 − 1.52i)35-s + (4.38 + 4.38i)37-s − 9.27i·41-s + (1.61 + 1.61i)43-s + 2.82·47-s + ⋯
L(s)  = 1  + (0.149 + 0.149i)5-s − 1.72i·7-s + (−0.745 − 0.745i)11-s + (−0.0164 + 0.0164i)13-s − 0.877·17-s + (−0.586 + 0.586i)19-s − 0.589i·23-s − 0.955i·25-s + (0.958 − 0.958i)29-s + 0.100·31-s + (0.258 − 0.258i)35-s + (0.721 + 0.721i)37-s − 1.44i·41-s + (0.245 + 0.245i)43-s + 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.154 + 0.987i$
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: \((2,\ 576,\ (\ :1/2),\ -0.154 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749396 - 0.875876i\)
\(L(\frac12)\) \(\approx\) \(0.749396 - 0.875876i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.334 - 0.334i)T + 5iT^{2} \)
7 \( 1 + 4.55iT - 7T^{2} \)
11 \( 1 + (2.47 + 2.47i)T + 11iT^{2} \)
13 \( 1 + (0.0594 - 0.0594i)T - 13iT^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
19 \( 1 + (2.55 - 2.55i)T - 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (-5.16 + 5.16i)T - 29iT^{2} \)
31 \( 1 - 0.557T + 31T^{2} \)
37 \( 1 + (-4.38 - 4.38i)T + 37iT^{2} \)
41 \( 1 + 9.27iT - 41T^{2} \)
43 \( 1 + (-1.61 - 1.61i)T + 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (-0.493 - 0.493i)T + 53iT^{2} \)
59 \( 1 + (-4 - 4i)T + 59iT^{2} \)
61 \( 1 + (-2.72 + 2.72i)T - 61iT^{2} \)
67 \( 1 + (3.77 - 3.77i)T - 67iT^{2} \)
71 \( 1 + 9.11iT - 71T^{2} \)
73 \( 1 - 0.541iT - 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + (10.6 - 10.6i)T - 83iT^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
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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

−10.53674204917505515465211742216, −9.902236988993900297219366426487, −8.537752125981318735663594643532, −7.86388917682531788350743479826, −6.84341561122601087467685697858, −6.08446683422302200754878188379, −4.64659558221367702359858444890, −3.88201608724888892725686382369, −2.49055164261047814902178716652, −0.63536017239316297815878463937, 2.01189740602251190856633781880, 2.90896615334330355947730813311, 4.62581080167913373127355882037, 5.39115907916154851700983563375, 6.33225266671685503596920243707, 7.41643966386611745821710211678, 8.554716027323699469993891150312, 9.095323739909218693625269818247, 9.968411323550486801781325375095, 11.08034245134727140914536845127

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