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Properties

Label 2-864-288.13-c1-0-3
Degree $2$
Conductor $864$
Sign $-0.684 - 0.728i$
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  + (−0.457 − 1.33i)2-s + (−1.58 + 1.22i)4-s + (−0.388 + 2.95i)5-s + (−1.26 − 0.338i)7-s + (2.36 + 1.55i)8-s + (4.12 − 0.830i)10-s + (−1.37 + 1.78i)11-s + (3.12 − 2.39i)13-s + (0.125 + 1.84i)14-s + (1.00 − 3.87i)16-s − 1.90i·17-s + (−3.62 − 1.50i)19-s + (−2.99 − 5.14i)20-s + (3.02 + 1.01i)22-s + (−4.83 + 1.29i)23-s + ⋯
L(s)  = 1  + (−0.323 − 0.946i)2-s + (−0.790 + 0.612i)4-s + (−0.173 + 1.31i)5-s + (−0.477 − 0.127i)7-s + (0.835 + 0.549i)8-s + (1.30 − 0.262i)10-s + (−0.413 + 0.539i)11-s + (0.865 − 0.664i)13-s + (0.0334 + 0.493i)14-s + (0.250 − 0.968i)16-s − 0.462i·17-s + (−0.832 − 0.344i)19-s + (−0.670 − 1.14i)20-s + (0.643 + 0.216i)22-s + (−1.00 + 0.270i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.684 - 0.728i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.684 - 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0967857 + 0.223724i\)
\(L(\frac12)\) \(\approx\) \(0.0967857 + 0.223724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.457 + 1.33i)T \)
3 \( 1 \)
good5 \( 1 + (0.388 - 2.95i)T + (-4.82 - 1.29i)T^{2} \)
7 \( 1 + (1.26 + 0.338i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.37 - 1.78i)T + (-2.84 - 10.6i)T^{2} \)
13 \( 1 + (-3.12 + 2.39i)T + (3.36 - 12.5i)T^{2} \)
17 \( 1 + 1.90iT - 17T^{2} \)
19 \( 1 + (3.62 + 1.50i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.83 - 1.29i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (5.10 - 0.672i)T + (28.0 - 7.50i)T^{2} \)
31 \( 1 + (5.15 - 8.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.79 + 2.40i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (7.92 - 2.12i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.61 - 6.01i)T + (-11.1 - 41.5i)T^{2} \)
47 \( 1 + (-3.53 + 2.03i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.38 + 10.5i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.228 + 1.73i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (-2.69 + 0.355i)T + (58.9 - 15.7i)T^{2} \)
67 \( 1 + (6.06 + 7.91i)T + (-17.3 + 64.7i)T^{2} \)
71 \( 1 + (10.8 - 10.8i)T - 71iT^{2} \)
73 \( 1 + (8.05 + 8.05i)T + 73iT^{2} \)
79 \( 1 + (-4.83 + 2.79i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.173 - 1.31i)T + (-80.1 + 21.4i)T^{2} \)
89 \( 1 + (-2.84 + 2.84i)T - 89iT^{2} \)
97 \( 1 + (-4.30 - 7.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.45699105822056712203612047518, −9.979970748152654513928451182985, −8.962495599273981964514413795765, −7.986392487939711939250619915446, −7.20561715385574500279220876162, −6.28712880012659375525872156501, −4.98551606545448839330028005698, −3.66436070310132493780725699729, −3.10424771028558784097047967321, −1.93375014515748140907219792637, 0.12969606833149240000521166053, 1.67654710646858742415558609282, 3.84134662886429256054844135749, 4.52769322503082776962645830544, 5.80996543782109327771417596010, 6.10414091264630275052473434412, 7.44665030105713622239827942822, 8.332757186530882520990426042464, 8.778729559530530929535462053321, 9.536138633386663779715222223774

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