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Properties

Label 2-24e2-576.13-c1-0-14
Degree $2$
Conductor $576$
Sign $-0.998 - 0.0491i$
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  + (−1.35 − 0.406i)2-s + (0.905 + 1.47i)3-s + (1.66 + 1.10i)4-s + (−1.35 + 3.99i)5-s + (−0.626 − 2.36i)6-s + (0.639 − 0.833i)7-s + (−1.81 − 2.17i)8-s + (−1.35 + 2.67i)9-s + (3.45 − 4.85i)10-s + (−6.14 − 0.402i)11-s + (−0.113 + 3.46i)12-s + (−0.436 + 0.498i)13-s + (−1.20 + 0.868i)14-s + (−7.11 + 1.61i)15-s + (1.57 + 3.67i)16-s + (3.81 + 3.81i)17-s + ⋯
L(s)  = 1  + (−0.957 − 0.287i)2-s + (0.523 + 0.852i)3-s + (0.834 + 0.550i)4-s + (−0.605 + 1.78i)5-s + (−0.255 − 0.966i)6-s + (0.241 − 0.315i)7-s + (−0.641 − 0.767i)8-s + (−0.452 + 0.891i)9-s + (1.09 − 1.53i)10-s + (−1.85 − 0.121i)11-s + (−0.0329 + 0.999i)12-s + (−0.121 + 0.138i)13-s + (−0.322 + 0.232i)14-s + (−1.83 + 0.417i)15-s + (0.393 + 0.919i)16-s + (0.924 + 0.924i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.998 - 0.0491i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.998 - 0.0491i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0150855 + 0.613272i\)
\(L(\frac12)\) \(\approx\) \(0.0150855 + 0.613272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.406i)T \)
3 \( 1 + (-0.905 - 1.47i)T \)
good5 \( 1 + (1.35 - 3.99i)T + (-3.96 - 3.04i)T^{2} \)
7 \( 1 + (-0.639 + 0.833i)T + (-1.81 - 6.76i)T^{2} \)
11 \( 1 + (6.14 + 0.402i)T + (10.9 + 1.43i)T^{2} \)
13 \( 1 + (0.436 - 0.498i)T + (-1.69 - 12.8i)T^{2} \)
17 \( 1 + (-3.81 - 3.81i)T + 17iT^{2} \)
19 \( 1 + (-0.464 + 2.33i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (-3.00 + 2.30i)T + (5.95 - 22.2i)T^{2} \)
29 \( 1 + (-2.11 + 4.29i)T + (-17.6 - 23.0i)T^{2} \)
31 \( 1 + (1.86 + 1.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.122 - 0.615i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-1.80 + 1.38i)T + (10.6 - 39.6i)T^{2} \)
43 \( 1 + (0.583 - 8.90i)T + (-42.6 - 5.61i)T^{2} \)
47 \( 1 + (-1.75 + 6.55i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.39 - 9.57i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (1.56 - 4.60i)T + (-46.8 - 35.9i)T^{2} \)
61 \( 1 + (-2.35 - 1.16i)T + (37.1 + 48.3i)T^{2} \)
67 \( 1 + (0.182 + 2.77i)T + (-66.4 + 8.74i)T^{2} \)
71 \( 1 + (-3.32 - 8.02i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.27 - 7.91i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.64 - 0.439i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (12.6 - 4.28i)T + (65.8 - 50.5i)T^{2} \)
89 \( 1 + (13.7 - 5.70i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (6.87 - 3.96i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.86733099920821745285557329975, −10.34446938631802109337168217609, −9.753505467784423698009440093855, −8.327862674603559264547040442805, −7.81104076236582871448751872478, −7.13077942017824311838703795344, −5.82367373041739584509954167704, −4.18467031559438206364801597494, −3.02731600144281796062178703942, −2.56858492000972404786718222961, 0.42751186461417399168170154747, 1.68255433871074129298376319345, 3.08267606943500495875806700745, 5.10768936448181339669792293298, 5.54596251436526127204251079787, 7.24668935012849278092874953088, 7.84989601570081002547363424556, 8.367771022531311052412586705124, 9.118242717665887536228454415004, 9.948205312105895685242241070193

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