Properties

Label 2-24e2-576.517-c1-0-2
Degree $2$
Conductor $576$
Sign $-0.933 + 0.358i$
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  + (−1.29 − 0.571i)2-s + (1.48 + 0.886i)3-s + (1.34 + 1.47i)4-s + (−2.64 + 3.02i)5-s + (−1.41 − 1.99i)6-s + (−4.44 + 0.585i)7-s + (−0.896 − 2.68i)8-s + (1.42 + 2.63i)9-s + (5.15 − 2.39i)10-s + (−0.255 + 0.518i)11-s + (0.691 + 3.39i)12-s + (2.16 − 6.37i)13-s + (6.08 + 1.78i)14-s + (−6.62 + 2.14i)15-s + (−0.374 + 3.98i)16-s + (−1.08 + 1.08i)17-s + ⋯
L(s)  = 1  + (−0.914 − 0.404i)2-s + (0.859 + 0.511i)3-s + (0.673 + 0.739i)4-s + (−1.18 + 1.35i)5-s + (−0.578 − 0.815i)6-s + (−1.68 + 0.221i)7-s + (−0.316 − 0.948i)8-s + (0.475 + 0.879i)9-s + (1.63 − 0.756i)10-s + (−0.0770 + 0.156i)11-s + (0.199 + 0.979i)12-s + (0.600 − 1.76i)13-s + (1.62 + 0.476i)14-s + (−1.70 + 0.554i)15-s + (−0.0936 + 0.995i)16-s + (−0.263 + 0.263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.933 + 0.358i$
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: \((2,\ 576,\ (\ :1/2),\ -0.933 + 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0312755 - 0.168923i\)
\(L(\frac12)\) \(\approx\) \(0.0312755 - 0.168923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.571i)T \)
3 \( 1 + (-1.48 - 0.886i)T \)
good5 \( 1 + (2.64 - 3.02i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (4.44 - 0.585i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (0.255 - 0.518i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (-2.16 + 6.37i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (1.08 - 1.08i)T - 17iT^{2} \)
19 \( 1 + (0.700 + 3.51i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (0.182 - 1.38i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (4.44 + 0.291i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (1.28 + 0.739i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.465 - 2.34i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (0.0670 - 0.509i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (-2.60 - 1.28i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (5.59 + 1.49i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.69 + 5.52i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (6.54 - 7.45i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (0.792 - 12.0i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (13.4 - 6.63i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (-0.626 + 1.51i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.95 + 11.9i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.658 - 2.45i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (6.13 - 5.37i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (-5.30 - 2.19i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-3.62 + 2.09i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.72662469406441891883493004813, −10.43017929950608066939483248272, −9.555175743104606680218865169459, −8.643631803832584912467429051673, −7.74730209851031201672993963766, −7.13603826182777353278544128922, −6.12997611810158575435875472198, −3.92275992354411795678107989349, −3.16661906482073579156339506123, −2.76619479242865612383417339583, 0.11450221508492032748063659455, 1.61725316419805756315533051127, 3.38283192809081015463652687270, 4.33515418663912155442545691535, 6.07860138557830280101302036030, 6.89972183960626468120562750713, 7.68456727247692695844252490795, 8.600486583719646451719784122904, 9.173454550927640519275752541787, 9.649341236347437348357463170503

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