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Properties

Label 2-24e2-48.11-c1-0-5
Degree $2$
Conductor $576$
Sign $0.873 + 0.487i$
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  + (2.10 − 2.10i)5-s + 4.40·7-s + (0.215 + 0.215i)11-s + (−2.73 + 2.73i)13-s − 2.36i·17-s + (−0.758 − 0.758i)19-s − 1.75i·23-s − 3.86i·25-s + (−5.54 − 5.54i)29-s + 9.01i·31-s + (9.27 − 9.27i)35-s + (3.10 + 3.10i)37-s + 10.1·41-s + (3.54 − 3.54i)43-s − 3.90·47-s + ⋯
L(s)  = 1  + (0.941 − 0.941i)5-s + 1.66·7-s + (0.0650 + 0.0650i)11-s + (−0.758 + 0.758i)13-s − 0.573i·17-s + (−0.174 − 0.174i)19-s − 0.366i·23-s − 0.772i·25-s + (−1.02 − 1.02i)29-s + 1.61i·31-s + (1.56 − 1.56i)35-s + (0.510 + 0.510i)37-s + 1.57·41-s + (0.540 − 0.540i)43-s − 0.569·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.873 + 0.487i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.873 + 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88450 - 0.489928i\)
\(L(\frac12)\) \(\approx\) \(1.88450 - 0.489928i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (-2.10 + 2.10i)T - 5iT^{2} \)
7 \( 1 - 4.40T + 7T^{2} \)
11 \( 1 + (-0.215 - 0.215i)T + 11iT^{2} \)
13 \( 1 + (2.73 - 2.73i)T - 13iT^{2} \)
17 \( 1 + 2.36iT - 17T^{2} \)
19 \( 1 + (0.758 + 0.758i)T + 19iT^{2} \)
23 \( 1 + 1.75iT - 23T^{2} \)
29 \( 1 + (5.54 + 5.54i)T + 29iT^{2} \)
31 \( 1 - 9.01iT - 31T^{2} \)
37 \( 1 + (-3.10 - 3.10i)T + 37iT^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + (-3.54 + 3.54i)T - 43iT^{2} \)
47 \( 1 + 3.90T + 47T^{2} \)
53 \( 1 + (2.71 - 2.71i)T - 53iT^{2} \)
59 \( 1 + (3.40 + 3.40i)T + 59iT^{2} \)
61 \( 1 + (1.75 - 1.75i)T - 61iT^{2} \)
67 \( 1 + (9.11 + 9.11i)T + 67iT^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 - 0.482iT - 73T^{2} \)
79 \( 1 + 6.88iT - 79T^{2} \)
83 \( 1 + (4.79 - 4.79i)T - 83iT^{2} \)
89 \( 1 + 7.00T + 89T^{2} \)
97 \( 1 + 3.34T + 97T^{2} \)
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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

−10.70611840978960830565200104117, −9.569417253224450972421454004829, −9.002782561053710512093669656763, −8.086492125976866639665898166719, −7.18023244958612805860203132748, −5.86047180741636313874345898925, −4.94570861435026903210127786005, −4.42686872334497919540388032797, −2.33519404220302742569774213635, −1.37376911664261722388456001293, 1.69673372608633264680205350187, 2.69027215142514935129637634571, 4.23193876431579066226859269572, 5.40055008816917425287042354434, 6.07589520638682967801534718555, 7.42065634095585160934114676668, 7.919784583050892030423842399711, 9.141620550988433742103693999575, 10.03608691005045257038406237448, 10.89785396271440221767530125666

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