Properties

Label 2-24e2-576.517-c1-0-23
Degree $2$
Conductor $576$
Sign $0.770 - 0.637i$
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  + (−0.536 − 1.30i)2-s + (1.62 − 0.594i)3-s + (−1.42 + 1.40i)4-s + (−1.11 + 1.27i)5-s + (−1.65 − 1.80i)6-s + (−2.38 + 0.314i)7-s + (2.60 + 1.10i)8-s + (2.29 − 1.93i)9-s + (2.27 + 0.779i)10-s + (−2.80 + 5.69i)11-s + (−1.48 + 3.13i)12-s + (−0.870 + 2.56i)13-s + (1.69 + 2.95i)14-s + (−1.06 + 2.74i)15-s + (0.0537 − 3.99i)16-s + (2.76 − 2.76i)17-s + ⋯
L(s)  = 1  + (−0.379 − 0.925i)2-s + (0.939 − 0.343i)3-s + (−0.711 + 0.702i)4-s + (−0.500 + 0.571i)5-s + (−0.674 − 0.738i)6-s + (−0.901 + 0.118i)7-s + (0.919 + 0.391i)8-s + (0.764 − 0.645i)9-s + (0.718 + 0.246i)10-s + (−0.846 + 1.71i)11-s + (−0.427 + 0.904i)12-s + (−0.241 + 0.711i)13-s + (0.452 + 0.789i)14-s + (−0.274 + 0.708i)15-s + (0.0134 − 0.999i)16-s + (0.670 − 0.670i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.770 - 0.637i$
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: \((2,\ 576,\ (\ :1/2),\ 0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.919056 + 0.331213i\)
\(L(\frac12)\) \(\approx\) \(0.919056 + 0.331213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.536 + 1.30i)T \)
3 \( 1 + (-1.62 + 0.594i)T \)
good5 \( 1 + (1.11 - 1.27i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (2.38 - 0.314i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (2.80 - 5.69i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (0.870 - 2.56i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (-2.76 + 2.76i)T - 17iT^{2} \)
19 \( 1 + (-1.19 - 5.98i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-0.265 + 2.01i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (7.63 + 0.500i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (-6.62 - 3.82i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.71 - 8.61i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-0.539 + 4.10i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (-5.33 - 2.63i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (4.40 + 1.18i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.39 + 3.58i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (5.68 - 6.48i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (-0.678 + 10.3i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (-11.2 + 5.55i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (-1.14 + 2.76i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.73 + 9.00i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (3.75 - 13.9i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-2.32 + 2.03i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (-14.7 - 6.12i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (7.37 - 4.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.58825640936182301306989036090, −9.655421510754054741106056379478, −9.566570842954933147428513930370, −8.110261525914140062572723026616, −7.51172078481705243570801222152, −6.75376281347704253796473693865, −4.84486369047569714562084073688, −3.68168400906472501382783488982, −2.89463766624719013515594220166, −1.82420225578305853958824217730, 0.56313793670037373973767696280, 2.95125634812809960640285294547, 3.94740839156662367908084621672, 5.18559691416073242126027766101, 6.04899667122767084551321903445, 7.40691049817635515127963568554, 8.032462349520156976210408973036, 8.721808392937432166466455282108, 9.478711806059963481315433091240, 10.32044789455088412725352060592

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