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Properties

Label 2-864-288.229-c1-0-28
Degree $2$
Conductor $864$
Sign $0.0332 + 0.999i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
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  + (−0.830 − 1.14i)2-s + (−0.619 + 1.90i)4-s + (1.83 − 1.41i)5-s + (0.454 − 1.69i)7-s + (2.69 − 0.870i)8-s + (−3.14 − 0.931i)10-s + (4.25 − 0.560i)11-s + (−0.303 + 2.30i)13-s + (−2.32 + 0.889i)14-s + (−3.23 − 2.35i)16-s − 1.85i·17-s + (2.22 − 0.920i)19-s + (1.54 + 4.36i)20-s + (−4.17 − 4.40i)22-s + (1.01 + 3.80i)23-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.309 + 0.950i)4-s + (0.821 − 0.630i)5-s + (0.171 − 0.641i)7-s + (0.951 − 0.307i)8-s + (−0.993 − 0.294i)10-s + (1.28 − 0.168i)11-s + (−0.0840 + 0.638i)13-s + (−0.620 + 0.237i)14-s + (−0.808 − 0.589i)16-s − 0.450i·17-s + (0.509 − 0.211i)19-s + (0.344 + 0.976i)20-s + (−0.890 − 0.938i)22-s + (0.212 + 0.793i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.0332 + 0.999i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.0332 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02959 - 0.995900i\)
\(L(\frac12)\) \(\approx\) \(1.02959 - 0.995900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.830 + 1.14i)T \)
3 \( 1 \)
good5 \( 1 + (-1.83 + 1.41i)T + (1.29 - 4.82i)T^{2} \)
7 \( 1 + (-0.454 + 1.69i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.25 + 0.560i)T + (10.6 - 2.84i)T^{2} \)
13 \( 1 + (0.303 - 2.30i)T + (-12.5 - 3.36i)T^{2} \)
17 \( 1 + 1.85iT - 17T^{2} \)
19 \( 1 + (-2.22 + 0.920i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.01 - 3.80i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.06 + 3.98i)T + (-7.50 - 28.0i)T^{2} \)
31 \( 1 + (3.53 - 6.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.25 - 1.34i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.20 + 11.9i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.39 - 0.315i)T + (41.5 - 11.1i)T^{2} \)
47 \( 1 + (-5.58 + 3.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.40 + 8.21i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (11.8 - 9.07i)T + (15.2 - 56.9i)T^{2} \)
61 \( 1 + (-8.00 + 10.4i)T + (-15.7 - 58.9i)T^{2} \)
67 \( 1 + (5.13 + 0.675i)T + (64.7 + 17.3i)T^{2} \)
71 \( 1 + (0.786 + 0.786i)T + 71iT^{2} \)
73 \( 1 + (-6.89 + 6.89i)T - 73iT^{2} \)
79 \( 1 + (7.90 - 4.56i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.78 + 2.13i)T + (21.4 + 80.1i)T^{2} \)
89 \( 1 + (3.68 + 3.68i)T + 89iT^{2} \)
97 \( 1 + (4.33 + 7.50i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.827892458230317315990647253394, −9.202297317861889022670278116410, −8.710955503066455282044633407866, −7.45705542947142759986235634828, −6.74732616305061926337466821012, −5.39305933898250470264940485554, −4.36094711618435590256535687919, −3.44017615097398616791613830489, −1.92963838005888630495259054059, −1.02125412262477356739300685916, 1.38325847070093417847722886318, 2.65828770164717373909651458201, 4.26999757696129671001032293215, 5.48465807495317110224491708996, 6.16481772192113540543171395913, 6.81191687248818409409885851009, 7.84333481044638381634425101426, 8.741137571661915856514610541227, 9.453981836177564732570535851384, 10.13297031295624854513565827294

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