Properties

Label 2-24e2-576.205-c1-0-68
Degree $2$
Conductor $576$
Sign $0.0198 + 0.999i$
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  + (−1.19 + 0.751i)2-s + (0.809 − 1.53i)3-s + (0.871 − 1.80i)4-s + (2.13 + 2.43i)5-s + (0.180 + 2.44i)6-s + (−3.26 − 0.430i)7-s + (0.307 + 2.81i)8-s + (−1.68 − 2.47i)9-s + (−4.38 − 1.31i)10-s + (−2.09 − 4.24i)11-s + (−2.05 − 2.79i)12-s + (−1.82 − 5.37i)13-s + (4.23 − 1.93i)14-s + (5.44 − 1.29i)15-s + (−2.48 − 3.13i)16-s + (0.816 + 0.816i)17-s + ⋯
L(s)  = 1  + (−0.847 + 0.531i)2-s + (0.467 − 0.884i)3-s + (0.435 − 0.900i)4-s + (0.953 + 1.08i)5-s + (0.0735 + 0.997i)6-s + (−1.23 − 0.162i)7-s + (0.108 + 0.994i)8-s + (−0.563 − 0.826i)9-s + (−1.38 − 0.414i)10-s + (−0.631 − 1.28i)11-s + (−0.592 − 0.805i)12-s + (−0.506 − 1.49i)13-s + (1.13 − 0.518i)14-s + (1.40 − 0.334i)15-s + (−0.620 − 0.784i)16-s + (0.198 + 0.198i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0198 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0198 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.0198 + 0.999i$
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: \((2,\ 576,\ (\ :1/2),\ 0.0198 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.627297 - 0.614965i\)
\(L(\frac12)\) \(\approx\) \(0.627297 - 0.614965i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 - 0.751i)T \)
3 \( 1 + (-0.809 + 1.53i)T \)
good5 \( 1 + (-2.13 - 2.43i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (3.26 + 0.430i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (2.09 + 4.24i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (1.82 + 5.37i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (-0.816 - 0.816i)T + 17iT^{2} \)
19 \( 1 + (0.428 - 2.15i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (1.06 + 8.08i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (-5.53 + 0.362i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (-7.01 + 4.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.116 - 0.587i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-0.609 - 4.63i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (6.66 - 3.28i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (-5.32 + 1.42i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.17 + 6.25i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (5.90 + 6.73i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (-0.257 - 3.93i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (1.94 + 0.958i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (-5.13 - 12.3i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.154 - 0.374i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (0.526 + 1.96i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-6.72 - 5.89i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (7.71 - 3.19i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (3.40 + 1.96i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.12118680473430110796377171250, −9.881238483011553613942857519684, −8.446451368803262568016079677366, −7.984524606557167034452100212444, −6.74511572701089856241233252081, −6.30793198187049561547783332058, −5.64435730731621074070468507196, −2.99710217548542918951997725686, −2.59918896884164830407127550097, −0.58302666701590794866822439656, 1.85633297393082944841770605814, 2.85812734715796308330348840935, 4.25230811330811123211183297962, 5.15797611116527298184761647093, 6.57267028093922571256311257622, 7.61657395905152855743407670072, 8.933074710241845774768079903230, 9.270865810305890588538157495269, 9.874440873183837524482675943563, 10.38243286048667865455505638740

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