Properties

Label 2-24e2-576.11-c1-0-20
Degree $2$
Conductor $576$
Sign $0.943 - 0.330i$
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  + (−1.28 + 0.592i)2-s + (−1.57 − 0.730i)3-s + (1.29 − 1.52i)4-s + (3.02 + 1.49i)5-s + (2.44 + 0.00813i)6-s + (−2.08 − 2.71i)7-s + (−0.766 + 2.72i)8-s + (1.93 + 2.29i)9-s + (−4.76 − 0.124i)10-s + (2.53 + 2.22i)11-s + (−3.15 + 1.44i)12-s + (3.20 + 0.210i)13-s + (4.27 + 2.25i)14-s + (−3.66 − 4.55i)15-s + (−0.628 − 3.95i)16-s + (−2.03 − 2.03i)17-s + ⋯
L(s)  = 1  + (−0.908 + 0.418i)2-s + (−0.906 − 0.421i)3-s + (0.649 − 0.760i)4-s + (1.35 + 0.667i)5-s + (0.999 + 0.00332i)6-s + (−0.786 − 1.02i)7-s + (−0.270 + 0.962i)8-s + (0.644 + 0.764i)9-s + (−1.50 − 0.0392i)10-s + (0.764 + 0.670i)11-s + (−0.909 + 0.415i)12-s + (0.889 + 0.0583i)13-s + (1.14 + 0.601i)14-s + (−0.945 − 1.17i)15-s + (−0.157 − 0.987i)16-s + (−0.494 − 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.943 - 0.330i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.943 - 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.879876 + 0.149459i\)
\(L(\frac12)\) \(\approx\) \(0.879876 + 0.149459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 0.592i)T \)
3 \( 1 + (1.57 + 0.730i)T \)
good5 \( 1 + (-3.02 - 1.49i)T + (3.04 + 3.96i)T^{2} \)
7 \( 1 + (2.08 + 2.71i)T + (-1.81 + 6.76i)T^{2} \)
11 \( 1 + (-2.53 - 2.22i)T + (1.43 + 10.9i)T^{2} \)
13 \( 1 + (-3.20 - 0.210i)T + (12.8 + 1.69i)T^{2} \)
17 \( 1 + (2.03 + 2.03i)T + 17iT^{2} \)
19 \( 1 + (4.27 - 6.40i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (-2.93 + 3.82i)T + (-5.95 - 22.2i)T^{2} \)
29 \( 1 + (-0.906 + 2.67i)T + (-23.0 - 17.6i)T^{2} \)
31 \( 1 + (-0.134 - 0.232i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.57 + 4.39i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-6.75 - 5.18i)T + (10.6 + 39.6i)T^{2} \)
43 \( 1 + (-5.98 - 5.24i)T + (5.61 + 42.6i)T^{2} \)
47 \( 1 + (-2.38 - 8.89i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.139 + 0.0278i)T + (48.9 + 20.2i)T^{2} \)
59 \( 1 + (-2.66 + 5.39i)T + (-35.9 - 46.8i)T^{2} \)
61 \( 1 + (1.29 - 3.82i)T + (-48.3 - 37.1i)T^{2} \)
67 \( 1 + (-8.74 + 7.66i)T + (8.74 - 66.4i)T^{2} \)
71 \( 1 + (-14.1 - 5.86i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (6.12 - 2.53i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (2.09 + 7.81i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-9.82 + 4.84i)T + (50.5 - 65.8i)T^{2} \)
89 \( 1 + (-5.49 - 2.27i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (12.0 + 6.93i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.72248567118205227013790503732, −9.871126028207069853549361392296, −9.374671485103794044691520179562, −7.906637501740472163978222820804, −6.87782259506598247633755716782, −6.36837542306002313250337011669, −5.92874928335304677341463832628, −4.31914914267033017956440781585, −2.33158775038890670755650720097, −1.11393236955372784416764663857, 0.990900262125243114099605187893, 2.42317743450173155964528083384, 3.88280485298467423719706222969, 5.40908808025252539831945685706, 6.21352745646544848956723453359, 6.71422297022022049094148275476, 8.717066021156067726133565834441, 9.058305885930396409725003064833, 9.619583316625183149436008949682, 10.70283678480549451315533066778

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