Properties

Label 2-24e2-16.5-c1-0-3
Degree $2$
Conductor $576$
Sign $0.857 - 0.513i$
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  + (2.68 + 2.68i)5-s − 2.15i·7-s + (1.79 + 1.79i)11-s + (1.38 − 1.38i)13-s + 0.224·17-s + (−0.158 + 0.158i)19-s + 2.82i·23-s + 9.42i·25-s + (1.85 − 1.85i)29-s − 1.84·31-s + (5.79 − 5.79i)35-s + (−3.66 − 3.66i)37-s + 5.88i·41-s + (7.75 + 7.75i)43-s − 2.82·47-s + ⋯
L(s)  = 1  + (1.20 + 1.20i)5-s − 0.816i·7-s + (0.542 + 0.542i)11-s + (0.383 − 0.383i)13-s + 0.0545·17-s + (−0.0364 + 0.0364i)19-s + 0.589i·23-s + 1.88i·25-s + (0.344 − 0.344i)29-s − 0.330·31-s + (0.980 − 0.980i)35-s + (−0.603 − 0.603i)37-s + 0.918i·41-s + (1.18 + 1.18i)43-s − 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.857 - 0.513i$
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: \((2,\ 576,\ (\ :1/2),\ 0.857 - 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77583 + 0.491166i\)
\(L(\frac12)\) \(\approx\) \(1.77583 + 0.491166i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.68 - 2.68i)T + 5iT^{2} \)
7 \( 1 + 2.15iT - 7T^{2} \)
11 \( 1 + (-1.79 - 1.79i)T + 11iT^{2} \)
13 \( 1 + (-1.38 + 1.38i)T - 13iT^{2} \)
17 \( 1 - 0.224T + 17T^{2} \)
19 \( 1 + (0.158 - 0.158i)T - 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (-1.85 + 1.85i)T - 29iT^{2} \)
31 \( 1 + 1.84T + 31T^{2} \)
37 \( 1 + (3.66 + 3.66i)T + 37iT^{2} \)
41 \( 1 - 5.88iT - 41T^{2} \)
43 \( 1 + (-7.75 - 7.75i)T + 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (7.51 + 7.51i)T + 53iT^{2} \)
59 \( 1 + (-4 - 4i)T + 59iT^{2} \)
61 \( 1 + (-5.98 + 5.98i)T - 61iT^{2} \)
67 \( 1 + (-10.4 + 10.4i)T - 67iT^{2} \)
71 \( 1 + 4.31iT - 71T^{2} \)
73 \( 1 + 5.97iT - 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 - 1.42iT - 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−10.75504517686956475488330532863, −9.902463128647903869554604021566, −9.427558186801703624502929612273, −7.995357825474630104077578471236, −7.01020406384434705920221386370, −6.40917011160616741632578321338, −5.42987564563612803745623732534, −4.01835546503428154920815679906, −2.90112530336452409137645105546, −1.60373925178502971529313638343, 1.27433624356110501397857124153, 2.47111821657142522495418113066, 4.10993928216977626968194442034, 5.31286622834080881661701657293, 5.85013110827075233514784414324, 6.84590380981995130880442894990, 8.504684374414577117546673401203, 8.812172047346572737440575556180, 9.575651569658471797224920473370, 10.52766243624315980537546620797

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