Properties

Label 2-24e2-576.517-c1-0-36
Degree $2$
Conductor $576$
Sign $0.0706 - 0.997i$
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  + (1.41 + 0.0725i)2-s + (−1.51 + 0.839i)3-s + (1.98 + 0.204i)4-s + (−1.38 + 1.57i)5-s + (−2.20 + 1.07i)6-s + (2.49 − 0.328i)7-s + (2.79 + 0.433i)8-s + (1.59 − 2.54i)9-s + (−2.06 + 2.12i)10-s + (−1.07 + 2.18i)11-s + (−3.18 + 1.35i)12-s + (−1.11 + 3.28i)13-s + (3.54 − 0.282i)14-s + (0.770 − 3.54i)15-s + (3.91 + 0.815i)16-s + (−1.22 + 1.22i)17-s + ⋯
L(s)  = 1  + (0.998 + 0.0513i)2-s + (−0.874 + 0.484i)3-s + (0.994 + 0.102i)4-s + (−0.617 + 0.703i)5-s + (−0.898 + 0.439i)6-s + (0.941 − 0.124i)7-s + (0.988 + 0.153i)8-s + (0.530 − 0.847i)9-s + (−0.652 + 0.671i)10-s + (−0.324 + 0.658i)11-s + (−0.919 + 0.392i)12-s + (−0.309 + 0.912i)13-s + (0.947 − 0.0755i)14-s + (0.198 − 0.914i)15-s + (0.978 + 0.203i)16-s + (−0.297 + 0.297i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0706 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0706 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.0706 - 0.997i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.0706 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44685 + 1.34804i\)
\(L(\frac12)\) \(\approx\) \(1.44685 + 1.34804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0725i)T \)
3 \( 1 + (1.51 - 0.839i)T \)
good5 \( 1 + (1.38 - 1.57i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (-2.49 + 0.328i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (1.07 - 2.18i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (1.11 - 3.28i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (1.22 - 1.22i)T - 17iT^{2} \)
19 \( 1 + (-0.214 - 1.07i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (0.00305 - 0.0232i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (7.22 + 0.473i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (-7.56 - 4.36i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.63 + 8.22i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-0.138 + 1.05i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (-1.20 - 0.596i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (-1.62 - 0.434i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.30 + 1.95i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-2.72 + 3.10i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (0.179 - 2.73i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (-13.8 + 6.82i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (-3.18 + 7.69i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.78 + 13.9i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-2.42 + 9.04i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-6.39 + 5.60i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (-11.5 - 4.79i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-1.38 + 0.802i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.00366892745929418420741838545, −10.61589661005638999853651697902, −9.418117007314417459384800452365, −7.84642482007605524499670240413, −7.14567209058830520024973710228, −6.31093324194928572707105442815, −5.15064989398410420563201951551, −4.45199203021415756255450083258, −3.59052675263793859745606164299, −1.95047097630436314861705948764, 0.960064782983126362465183316986, 2.53457302570458578308134165258, 4.16613675530321378411795008426, 5.03106816638608299474114611289, 5.59963508023026050510928931454, 6.72435233437872867050237269719, 7.84323389264381617913872759918, 8.234247322723618397578988917257, 10.00025522144043543492424418081, 11.04027605730352748548400641356

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