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Properties

Label 2-24e2-72.61-c1-0-1
Degree $2$
Conductor $576$
Sign $0.189 - 0.981i$
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  + (−1.59 − 0.675i)3-s + (1.5 − 0.866i)5-s + (−1.80 + 3.12i)7-s + (2.08 + 2.15i)9-s + (−0.635 − 0.367i)11-s + (−0.527 + 0.304i)13-s + (−2.97 + 0.367i)15-s − 5.52·17-s + 2i·19-s + (4.99 − 3.76i)21-s + (2.36 + 4.10i)23-s + (−1 + 1.73i)25-s + (−1.86 − 4.84i)27-s + (6.78 + 3.91i)29-s + (4.70 + 8.15i)31-s + ⋯
L(s)  = 1  + (−0.920 − 0.390i)3-s + (0.670 − 0.387i)5-s + (−0.682 + 1.18i)7-s + (0.695 + 0.718i)9-s + (−0.191 − 0.110i)11-s + (−0.146 + 0.0845i)13-s + (−0.768 + 0.0947i)15-s − 1.33·17-s + 0.458i·19-s + (1.09 − 0.822i)21-s + (0.493 + 0.855i)23-s + (−0.200 + 0.346i)25-s + (−0.359 − 0.933i)27-s + (1.26 + 0.727i)29-s + (0.845 + 1.46i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.633358 + 0.522631i\)
\(L(\frac12)\) \(\approx\) \(0.633358 + 0.522631i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 + 0.675i)T \)
good5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.80 - 3.12i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.635 + 0.367i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.527 - 0.304i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.52T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-2.36 - 4.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.78 - 3.91i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.70 - 8.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.34iT - 37T^{2} \)
41 \( 1 + (-4.26 - 7.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.88 + 5.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.88 + 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 13.0iT - 53T^{2} \)
59 \( 1 + (1.04 - 0.604i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.78 + 5.65i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.46 - 3.15i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.63T + 71T^{2} \)
73 \( 1 + 2.05T + 73T^{2} \)
79 \( 1 + (1.24 - 2.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.6 - 6.12i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.94T + 89T^{2} \)
97 \( 1 + (-7.78 + 13.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.93581251563214788284352219548, −10.07813244236770719926049932543, −9.185431293751918617922163242083, −8.432991928422463660935506836932, −7.02476753586561658953183028369, −6.28215683633664404616975869539, −5.50173659601771909531650719759, −4.71605478318418432339651592084, −2.88292870468792891226367333749, −1.59040847755227547342418417365, 0.52271847770359384192678164231, 2.55975494380091523602067502104, 4.08013241050640157863832172182, 4.80340818168965557594474213454, 6.32396662845050633581186921810, 6.50018472163679674786547559367, 7.62467514558388932089310181705, 9.088950888290091826736337068638, 9.964592116578513071853449980378, 10.46312028708803134523717627714

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