Properties

Label 2-24e2-576.517-c1-0-58
Degree $2$
Conductor $576$
Sign $0.227 - 0.973i$
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  + (1.06 + 0.931i)2-s + (1.31 + 1.13i)3-s + (0.264 + 1.98i)4-s + (2.27 − 2.59i)5-s + (0.342 + 2.42i)6-s + (−1.60 + 0.211i)7-s + (−1.56 + 2.35i)8-s + (0.441 + 2.96i)9-s + (4.83 − 0.640i)10-s + (1.80 − 3.65i)11-s + (−1.89 + 2.89i)12-s + (−0.973 + 2.86i)13-s + (−1.90 − 1.27i)14-s + (5.91 − 0.828i)15-s + (−3.86 + 1.04i)16-s + (1.07 − 1.07i)17-s + ⋯
L(s)  = 1  + (0.752 + 0.658i)2-s + (0.757 + 0.653i)3-s + (0.132 + 0.991i)4-s + (1.01 − 1.15i)5-s + (0.139 + 0.990i)6-s + (−0.606 + 0.0798i)7-s + (−0.553 + 0.832i)8-s + (0.147 + 0.989i)9-s + (1.52 − 0.202i)10-s + (0.543 − 1.10i)11-s + (−0.547 + 0.837i)12-s + (−0.270 + 0.795i)13-s + (−0.508 − 0.339i)14-s + (1.52 − 0.214i)15-s + (−0.965 + 0.262i)16-s + (0.261 − 0.261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.227 - 0.973i$
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: \((2,\ 576,\ (\ :1/2),\ 0.227 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36306 + 1.87551i\)
\(L(\frac12)\) \(\approx\) \(2.36306 + 1.87551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 - 0.931i)T \)
3 \( 1 + (-1.31 - 1.13i)T \)
good5 \( 1 + (-2.27 + 2.59i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (1.60 - 0.211i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (-1.80 + 3.65i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (0.973 - 2.86i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (-1.07 + 1.07i)T - 17iT^{2} \)
19 \( 1 + (-1.23 - 6.19i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-0.0343 + 0.260i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (-0.590 - 0.0387i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (6.42 + 3.70i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.25 + 11.3i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-1.59 + 12.1i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (-5.57 - 2.74i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (2.98 + 0.799i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.17 + 7.73i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (5.38 - 6.14i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (0.313 - 4.78i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (1.06 - 0.522i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (4.32 - 10.4i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (0.785 + 1.89i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.57 + 5.88i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (2.84 - 2.49i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (-16.7 - 6.95i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (6.95 - 4.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.88646691502091790497914947297, −9.545302418804459905225097823551, −9.179578179963621992287787311785, −8.409578683995438041989646857987, −7.35498601409870745261633510201, −5.94101878829380832039178229609, −5.52119697496680694393466647587, −4.28890407241168593739325267434, −3.47723884640923923288855372585, −2.05531450906780580706547721358, 1.59121110883276186673658974536, 2.74234585340186693228516151131, 3.26804242967800643050460443718, 4.81076815057699261477706384062, 6.23023333044319221720770916225, 6.66250856165533007043046061776, 7.55725539997057008398989892321, 9.273660695218721189473571173494, 9.709967452719626847885536789105, 10.46761057812311428815332162644

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