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Properties

Label 2-24e2-144.13-c1-0-16
Degree $2$
Conductor $576$
Sign $0.537 + 0.843i$
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  + (1.5 − 0.866i)3-s + (−0.267 − i)5-s + (2.36 − 1.36i)7-s + (1.5 − 2.59i)9-s + (4.23 + 1.13i)11-s + (−3.36 + 0.901i)13-s + (−1.26 − 1.26i)15-s − 5.73·17-s + (2.36 + 2.36i)19-s + (2.36 − 4.09i)21-s + (−4.09 − 2.36i)23-s + (3.40 − 1.96i)25-s − 5.19i·27-s + (−0.633 + 2.36i)29-s + (0.267 − 0.464i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.119 − 0.447i)5-s + (0.894 − 0.516i)7-s + (0.5 − 0.866i)9-s + (1.27 + 0.341i)11-s + (−0.933 + 0.250i)13-s + (−0.327 − 0.327i)15-s − 1.39·17-s + (0.542 + 0.542i)19-s + (0.516 − 0.894i)21-s + (−0.854 − 0.493i)23-s + (0.680 − 0.392i)25-s − 0.999i·27-s + (−0.117 + 0.439i)29-s + (0.0481 − 0.0833i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.537 + 0.843i$
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: \((2,\ 576,\ (\ :1/2),\ 0.537 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83283 - 1.00552i\)
\(L(\frac12)\) \(\approx\) \(1.83283 - 1.00552i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
good5 \( 1 + (0.267 + i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.36 + 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.23 - 1.13i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (3.36 - 0.901i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 + (-2.36 - 2.36i)T + 19iT^{2} \)
23 \( 1 + (4.09 + 2.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.633 - 2.36i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-0.267 + 0.464i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.73 + 4.73i)T - 37iT^{2} \)
41 \( 1 + (-2.59 - 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.33 - 2.23i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.83 + 6.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.46 - 7.46i)T - 53iT^{2} \)
59 \( 1 + (-1.96 - 7.33i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3 - 11.1i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (6.59 - 1.76i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 + 6.26iT - 73T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.366 - 1.36i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 + (5.86 + 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.53845161253055456786191031745, −9.394310191057611213834787191377, −8.881611924631788124660252003312, −7.87717361387222809149298613794, −7.20392190230465880518958012204, −6.26041863570242261577086910155, −4.58169251470589773119412884980, −4.07698259689205053550646436199, −2.40603293811004784736397263174, −1.26848366388194224536007203471, 1.91493704273533030447488103580, 3.01140495584972192352855143522, 4.22426791487876113833159390421, 5.04513515440582730632390454204, 6.44067288560019112536254488185, 7.47500259637307731378188261310, 8.297115184287792396439669398711, 9.179509816783846416751313992950, 9.726108973614955566295964446959, 11.04601386011717249705916517621

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