Properties

Label 2-24e2-576.205-c1-0-62
Degree $2$
Conductor $576$
Sign $0.211 - 0.977i$
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  + (1.26 + 0.640i)2-s + (1.37 + 1.05i)3-s + (1.18 + 1.61i)4-s + (0.347 + 0.396i)5-s + (1.05 + 2.20i)6-s + (2.56 + 0.338i)7-s + (0.454 + 2.79i)8-s + (0.773 + 2.89i)9-s + (0.184 + 0.723i)10-s + (−1.93 − 3.93i)11-s + (−0.0829 + 3.46i)12-s + (−1.46 − 4.31i)13-s + (3.02 + 2.07i)14-s + (0.0592 + 0.912i)15-s + (−1.21 + 3.81i)16-s + (−3.42 − 3.42i)17-s + ⋯
L(s)  = 1  + (0.891 + 0.452i)2-s + (0.793 + 0.609i)3-s + (0.590 + 0.807i)4-s + (0.155 + 0.177i)5-s + (0.431 + 0.902i)6-s + (0.970 + 0.127i)7-s + (0.160 + 0.987i)8-s + (0.257 + 0.966i)9-s + (0.0584 + 0.228i)10-s + (−0.584 − 1.18i)11-s + (−0.0239 + 0.999i)12-s + (−0.406 − 1.19i)13-s + (0.807 + 0.553i)14-s + (0.0153 + 0.235i)15-s + (−0.303 + 0.952i)16-s + (−0.831 − 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.55813 + 2.06370i\)
\(L(\frac12)\) \(\approx\) \(2.55813 + 2.06370i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.640i)T \)
3 \( 1 + (-1.37 - 1.05i)T \)
good5 \( 1 + (-0.347 - 0.396i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (-2.56 - 0.338i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (1.93 + 3.93i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (1.46 + 4.31i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (3.42 + 3.42i)T + 17iT^{2} \)
19 \( 1 + (0.675 - 3.39i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (0.716 + 5.44i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (2.67 - 0.175i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (3.23 - 1.86i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.61 - 8.12i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-1.30 - 9.91i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (-4.70 + 2.32i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (-6.57 + 1.76i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.45 + 3.67i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (5.61 + 6.40i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (-0.992 - 15.1i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (9.00 + 4.44i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (-3.11 - 7.52i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-4.52 + 10.9i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.461 - 1.72i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (10.8 + 9.49i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (1.48 - 0.616i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (-15.5 - 9.00i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.85574620988969062910163864202, −10.29299152593614599231839654326, −8.785373447243140567723128303445, −8.179244575248380522042532353548, −7.54995061554631899788983678906, −6.14742785290094009145841734273, −5.17417602316979721621680974645, −4.45431233358642520834850509662, −3.15515192365973845473219132848, −2.38762414996017348394184266489, 1.75020544229334320525919893697, 2.25096501287024878698213526278, 3.91460271390949273242319071063, 4.65852200568847761286338770321, 5.81333323837263794845491059129, 7.18562034435270113594758250460, 7.45188055520073715715346140722, 8.989429760957924153442350833756, 9.543885373914627042837626970804, 10.82323429265854713110885465421

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