Properties

Label 2-24e2-3.2-c6-0-0
Degree $2$
Conductor $576$
Sign $-0.577 - 0.816i$
Analytic cond. $132.511$
Root an. cond. $11.5113$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 103. i·5-s − 1.65e3·13-s − 6.21e3i·17-s + 4.96e3·25-s + 3.56e3i·29-s − 5.55e4·37-s + 1.67e4i·41-s − 1.17e5·49-s − 2.30e5i·53-s − 2.34e5·61-s + 1.70e5i·65-s + 6.50e5·73-s − 6.41e5·85-s + 7.67e5i·89-s + 1.07e6·97-s + ⋯
L(s)  = 1  − 0.825i·5-s − 0.753·13-s − 1.26i·17-s + 0.317·25-s + 0.146i·29-s − 1.09·37-s + 0.242i·41-s − 49-s − 1.54i·53-s − 1.03·61-s + 0.622i·65-s + 1.67·73-s − 1.04·85-s + 1.08i·89-s + 1.18·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(132.511\)
Root analytic conductor: \(11.5113\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.09329593060\)
\(L(\frac12)\) \(\approx\) \(0.09329593060\)
\(L(4)\) 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 \( 1 + 103. iT - 1.56e4T^{2} \)
7 \( 1 + 1.17e5T^{2} \)
11 \( 1 - 1.77e6T^{2} \)
13 \( 1 + 1.65e3T + 4.82e6T^{2} \)
17 \( 1 + 6.21e3iT - 2.41e7T^{2} \)
19 \( 1 + 4.70e7T^{2} \)
23 \( 1 - 1.48e8T^{2} \)
29 \( 1 - 3.56e3iT - 5.94e8T^{2} \)
31 \( 1 + 8.87e8T^{2} \)
37 \( 1 + 5.55e4T + 2.56e9T^{2} \)
41 \( 1 - 1.67e4iT - 4.75e9T^{2} \)
43 \( 1 + 6.32e9T^{2} \)
47 \( 1 - 1.07e10T^{2} \)
53 \( 1 + 2.30e5iT - 2.21e10T^{2} \)
59 \( 1 - 4.21e10T^{2} \)
61 \( 1 + 2.34e5T + 5.15e10T^{2} \)
67 \( 1 + 9.04e10T^{2} \)
71 \( 1 - 1.28e11T^{2} \)
73 \( 1 - 6.50e5T + 1.51e11T^{2} \)
79 \( 1 + 2.43e11T^{2} \)
83 \( 1 - 3.26e11T^{2} \)
89 \( 1 - 7.67e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.07e6T + 8.32e11T^{2} \)
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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

−9.883850238927567788295968170457, −9.236485369573695849615189280423, −8.377614774582335277076538103104, −7.43412565774805401372519800697, −6.54663847074108616080283997854, −5.19733825499674766625518509855, −4.78207319795807977791697792850, −3.42864118816827463727848090644, −2.25170215900847200047149413271, −1.00142957499697520391333921063, 0.02009874049664592055004052627, 1.58074804641242040579825958747, 2.68679126481278772277016208048, 3.66251410533916320432010938252, 4.79133552674380420816304260919, 5.93461729155541545814879542640, 6.77403057071346917611064601380, 7.60521057102904600263326165761, 8.541707784848133427601742590589, 9.561960090031519415400156754267

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