Properties

Label 2-24e2-576.205-c1-0-75
Degree $2$
Conductor $576$
Sign $-0.199 + 0.979i$
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  + (1.32 + 0.487i)2-s + (−1.45 + 0.933i)3-s + (1.52 + 1.29i)4-s + (−2.04 − 2.32i)5-s + (−2.39 + 0.527i)6-s + (−3.49 − 0.460i)7-s + (1.39 + 2.46i)8-s + (1.25 − 2.72i)9-s + (−1.57 − 4.08i)10-s + (0.915 + 1.85i)11-s + (−3.43 − 0.467i)12-s + (−0.864 − 2.54i)13-s + (−4.41 − 2.31i)14-s + (5.14 + 1.48i)15-s + (0.643 + 3.94i)16-s + (−5.16 − 5.16i)17-s + ⋯
L(s)  = 1  + (0.938 + 0.345i)2-s + (−0.842 + 0.539i)3-s + (0.761 + 0.647i)4-s + (−0.912 − 1.04i)5-s + (−0.976 + 0.215i)6-s + (−1.32 − 0.173i)7-s + (0.491 + 0.870i)8-s + (0.418 − 0.908i)9-s + (−0.497 − 1.29i)10-s + (0.276 + 0.559i)11-s + (−0.990 − 0.134i)12-s + (−0.239 − 0.706i)13-s + (−1.18 − 0.619i)14-s + (1.32 + 0.384i)15-s + (0.160 + 0.986i)16-s + (−1.25 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.199 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.199 + 0.979i$
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: \((2,\ 576,\ (\ :1/2),\ -0.199 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401840 - 0.491745i\)
\(L(\frac12)\) \(\approx\) \(0.401840 - 0.491745i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 - 0.487i)T \)
3 \( 1 + (1.45 - 0.933i)T \)
good5 \( 1 + (2.04 + 2.32i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (3.49 + 0.460i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (-0.915 - 1.85i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (0.864 + 2.54i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (5.16 + 5.16i)T + 17iT^{2} \)
19 \( 1 + (-1.60 + 8.07i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (-0.0532 - 0.404i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (3.08 - 0.201i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (7.84 - 4.52i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.953 + 4.79i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-0.508 - 3.86i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (-9.63 + 4.75i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (9.03 - 2.42i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.824 - 1.23i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (-7.05 - 8.03i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (-0.367 - 5.60i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (1.26 + 0.623i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (0.142 + 0.342i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-0.715 + 1.72i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.200 - 0.747i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (9.36 + 8.21i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (-0.00819 + 0.00339i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (-6.18 - 3.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.83665597844016142743893219310, −9.471151696763155389283452489104, −8.904207059249168666149480761968, −7.23168318666909620885213752046, −6.93950161906277660678266345786, −5.60932121131460916452999970952, −4.79094726964126461251760331277, −4.12346580540032359368058340071, −2.99674291911932494549741509136, −0.27581582226771441199578068057, 2.01454442303794339522086545925, 3.46649331430277526955756368667, 4.07804205496149235852849858444, 5.71948881603884073692254310588, 6.39301150172555870732155637142, 6.92363707570742848970376196595, 7.937817567312515523349416865224, 9.592287183209847606350511549224, 10.53768556829614988508860142565, 11.18495562175820991643595295706

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