Properties

Label 2-24e2-576.205-c1-0-82
Degree $2$
Conductor $576$
Sign $0.646 + 0.762i$
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  + (1.40 + 0.145i)2-s + (1.66 − 0.468i)3-s + (1.95 + 0.409i)4-s + (−2.43 − 2.77i)5-s + (2.41 − 0.416i)6-s + (−1.08 − 0.142i)7-s + (2.69 + 0.861i)8-s + (2.56 − 1.56i)9-s + (−3.02 − 4.26i)10-s + (−0.293 − 0.595i)11-s + (3.45 − 0.233i)12-s + (−1.92 − 5.65i)13-s + (−1.50 − 0.358i)14-s + (−5.36 − 3.48i)15-s + (3.66 + 1.60i)16-s + (5.07 + 5.07i)17-s + ⋯
L(s)  = 1  + (0.994 + 0.102i)2-s + (0.962 − 0.270i)3-s + (0.978 + 0.204i)4-s + (−1.08 − 1.24i)5-s + (0.985 − 0.169i)6-s + (−0.409 − 0.0538i)7-s + (0.952 + 0.304i)8-s + (0.853 − 0.520i)9-s + (−0.955 − 1.34i)10-s + (−0.0885 − 0.179i)11-s + (0.997 − 0.0675i)12-s + (−0.532 − 1.56i)13-s + (−0.401 − 0.0957i)14-s + (−1.38 − 0.901i)15-s + (0.916 + 0.401i)16-s + (1.22 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.646 + 0.762i$
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: \((2,\ 576,\ (\ :1/2),\ 0.646 + 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.78028 - 1.28760i\)
\(L(\frac12)\) \(\approx\) \(2.78028 - 1.28760i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.145i)T \)
3 \( 1 + (-1.66 + 0.468i)T \)
good5 \( 1 + (2.43 + 2.77i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (1.08 + 0.142i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (0.293 + 0.595i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (1.92 + 5.65i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (-5.07 - 5.07i)T + 17iT^{2} \)
19 \( 1 + (0.437 - 2.20i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (-0.588 - 4.46i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (-6.42 + 0.421i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (7.15 - 4.12i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.377 - 1.89i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-0.0252 - 0.192i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (8.29 - 4.08i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (-8.82 + 2.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.96 - 2.94i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (4.73 + 5.39i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (-0.133 - 2.04i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (-0.0718 - 0.0354i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (-1.19 - 2.89i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-1.57 + 3.79i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.51 - 5.66i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-2.74 - 2.40i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (-8.96 + 3.71i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (13.9 + 8.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.64279738389789537418947436854, −9.746963556906755874405017428537, −8.308395315214194001201840057615, −8.058158523185730847433441908232, −7.21565055993638998761912399322, −5.81004383197696015865267600604, −4.87763215577055574118816271160, −3.69346501459443147866223823292, −3.19291422989148779823122057033, −1.33648393647047179395407998266, 2.36806529638720380805215648523, 3.13624287586913804229873442936, 4.00903368144047570920793034473, 4.88173258513339565170350475086, 6.61197820179104057219861832595, 7.15383531381234008302041536228, 7.82138455575305761123670813610, 9.227471078988392010842481868192, 10.12518587330581539509750068904, 10.94993306261679166500589226817

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