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Properties

Label 2-24e2-144.11-c1-0-11
Degree $2$
Conductor $576$
Sign $0.977 + 0.212i$
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  + (−1.70 + 0.313i)3-s + (2.80 + 0.752i)5-s + (1.02 − 1.77i)7-s + (2.80 − 1.06i)9-s + (1.67 − 0.447i)11-s + (−4.86 − 1.30i)13-s + (−5.02 − 0.402i)15-s − 2.90i·17-s + (3.23 + 3.23i)19-s + (−1.19 + 3.34i)21-s + (0.831 − 0.480i)23-s + (2.99 + 1.72i)25-s + (−4.44 + 2.69i)27-s + (5.55 − 1.48i)29-s + (8.00 − 4.62i)31-s + ⋯
L(s)  = 1  + (−0.983 + 0.180i)3-s + (1.25 + 0.336i)5-s + (0.387 − 0.671i)7-s + (0.934 − 0.355i)9-s + (0.503 − 0.135i)11-s + (−1.34 − 0.361i)13-s + (−1.29 − 0.103i)15-s − 0.705i·17-s + (0.741 + 0.741i)19-s + (−0.259 + 0.730i)21-s + (0.173 − 0.100i)23-s + (0.599 + 0.345i)25-s + (−0.854 + 0.518i)27-s + (1.03 − 0.276i)29-s + (1.43 − 0.830i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.977 + 0.212i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.977 + 0.212i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38992 - 0.149550i\)
\(L(\frac12)\) \(\approx\) \(1.38992 - 0.149550i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 0.313i)T \)
good5 \( 1 + (-2.80 - 0.752i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.02 + 1.77i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.67 + 0.447i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (4.86 + 1.30i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 2.90iT - 17T^{2} \)
19 \( 1 + (-3.23 - 3.23i)T + 19iT^{2} \)
23 \( 1 + (-0.831 + 0.480i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.55 + 1.48i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-8.00 + 4.62i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.18 - 8.18i)T + 37iT^{2} \)
41 \( 1 + (4.06 + 7.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.907 + 3.38i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.60 + 2.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.65 - 1.65i)T - 53iT^{2} \)
59 \( 1 + (2.21 - 8.27i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.284 + 1.06i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.50 - 9.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.40iT - 71T^{2} \)
73 \( 1 - 0.312iT - 73T^{2} \)
79 \( 1 + (-1.15 - 0.669i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.90 + 14.5i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 7.86T + 89T^{2} \)
97 \( 1 + (6.48 - 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.34515400209086033821274677344, −10.12718253865854476460038062017, −9.347918960537822927078069200096, −7.80538669927412105513019552905, −6.91289393315252739196055370988, −6.09162456675560657248238275233, −5.20427157479526904020213253929, −4.33443994714852705939061372830, −2.65688000563914712446207702922, −1.09213467610074666218496119315, 1.36501931111224451182731518278, 2.51112535731779468761267293617, 4.63525796395829854915460379689, 5.18898360260715590097374893023, 6.15850014149264606630733765124, 6.86100637651255637831561059074, 8.080133540245986897185328566861, 9.350555064835878496533232871829, 9.762093925511820165402312581276, 10.75771255254432494643447641352

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