Properties

Label 2-24e2-144.13-c1-0-0
Degree $2$
Conductor $576$
Sign $-0.881 - 0.472i$
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  + (−0.241 − 1.71i)3-s + (0.531 + 1.98i)5-s + (−1.54 + 0.894i)7-s + (−2.88 + 0.828i)9-s + (−2.58 − 0.693i)11-s + (−4.63 + 1.24i)13-s + (3.27 − 1.39i)15-s − 3.58·17-s + (−4.85 − 4.85i)19-s + (1.90 + 2.44i)21-s + (0.446 + 0.257i)23-s + (0.675 − 0.390i)25-s + (2.11 + 4.74i)27-s + (1.72 − 6.44i)29-s + (−4.05 + 7.01i)31-s + ⋯
L(s)  = 1  + (−0.139 − 0.990i)3-s + (0.237 + 0.887i)5-s + (−0.585 + 0.338i)7-s + (−0.961 + 0.276i)9-s + (−0.779 − 0.208i)11-s + (−1.28 + 0.344i)13-s + (0.845 − 0.359i)15-s − 0.870·17-s + (−1.11 − 1.11i)19-s + (0.416 + 0.532i)21-s + (0.0930 + 0.0537i)23-s + (0.135 − 0.0780i)25-s + (0.407 + 0.913i)27-s + (0.320 − 1.19i)29-s + (−0.727 + 1.26i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.881 - 0.472i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.881 - 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0192816 + 0.0768352i\)
\(L(\frac12)\) \(\approx\) \(0.0192816 + 0.0768352i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.241 + 1.71i)T \)
good5 \( 1 + (-0.531 - 1.98i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.54 - 0.894i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.58 + 0.693i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (4.63 - 1.24i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 3.58T + 17T^{2} \)
19 \( 1 + (4.85 + 4.85i)T + 19iT^{2} \)
23 \( 1 + (-0.446 - 0.257i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.72 + 6.44i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (4.05 - 7.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.25 + 1.25i)T - 37iT^{2} \)
41 \( 1 + (-4.07 - 2.35i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.57 + 1.76i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.48 - 6.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.26 + 5.26i)T - 53iT^{2} \)
59 \( 1 + (-1.81 - 6.76i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.55 - 5.78i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.69 - 0.453i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.58iT - 71T^{2} \)
73 \( 1 - 12.5iT - 73T^{2} \)
79 \( 1 + (-4.01 - 6.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.14 + 7.99i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 16.5iT - 89T^{2} \)
97 \( 1 + (4.15 + 7.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.07123791395701496914632766815, −10.40040515759902929684698089641, −9.282718201153560275529129066683, −8.386100876053095028795853852315, −7.21872788453382652667599147452, −6.74758534296218445893592342326, −5.87936240574683787231760191803, −4.68132190693022780035236040218, −2.80323488622197019740360121648, −2.33957551428114660824972647079, 0.04073785117965483172370745987, 2.38007422906141270316635275823, 3.77289308092525181588963063991, 4.78785055939554138447494945359, 5.43684721881110107985685477267, 6.60345032347590917695935560608, 7.86186339467216952717044108811, 8.800613322828788386629884498499, 9.575352835138375849918286159075, 10.27892858768773141063884023326

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