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Properties

Label 2-24e2-144.13-c1-0-14
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  − 1.73i·3-s + (0.5 + 1.86i)5-s + (3.86 − 2.23i)7-s − 2.99·9-s + (−1.86 − 0.5i)11-s + (2.23 − 0.598i)13-s + (3.23 − 0.866i)15-s + 4·17-s + (3 + 3i)19-s + (−3.86 − 6.69i)21-s + (−5.59 − 3.23i)23-s + (1.09 − 0.633i)25-s + 5.19i·27-s + (0.232 − 0.866i)29-s + (4.59 − 7.96i)31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.223 + 0.834i)5-s + (1.46 − 0.843i)7-s − 0.999·9-s + (−0.562 − 0.150i)11-s + (0.619 − 0.165i)13-s + (0.834 − 0.223i)15-s + 0.970·17-s + (0.688 + 0.688i)19-s + (−0.843 − 1.46i)21-s + (−1.16 − 0.673i)23-s + (0.219 − 0.126i)25-s + 0.999i·27-s + (0.0430 − 0.160i)29-s + (0.825 − 1.43i)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

\(L(1)\) \(\approx\) \(1.50151 - 0.823757i\)
\(L(\frac12)\) \(\approx\) \(1.50151 - 0.823757i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
good5 \( 1 + (-0.5 - 1.86i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-3.86 + 2.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.86 + 0.5i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.23 + 0.598i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (-3 - 3i)T + 19iT^{2} \)
23 \( 1 + (5.59 + 3.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.232 + 0.866i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-4.59 + 7.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.26 - 4.26i)T - 37iT^{2} \)
41 \( 1 + (0.696 + 0.401i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.33 + 1.69i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.598 - 1.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.73 + 5.73i)T - 53iT^{2} \)
59 \( 1 + (0.401 + 1.5i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.571 + 2.13i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (8.33 - 2.23i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 - 7.46iT - 73T^{2} \)
79 \( 1 + (-0.866 - 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.79 - 14.1i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 15.8iT - 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.62509584298074654973293549855, −10.02576715115442846773954383502, −8.205764483471003760930020134329, −8.059214217030079325605654486350, −7.10188257370848706768414269321, −6.12887263880808359540798833493, −5.19019176032247911889527181993, −3.73008795082027087734488673489, −2.40543256818427037116487358503, −1.16025467023798581145218401595, 1.59734614208613301947217656165, 3.13318201204148951053041642960, 4.56378366603511762770671315633, 5.19796074638179476577494345650, 5.80376915092587888653365147576, 7.60861180980191805151075174702, 8.537811912240434605340389561913, 8.933370294478129481250885408174, 9.993312162001975866665113304352, 10.80602201788239206924527908033

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