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Properties

Label 2-864-288.13-c1-0-30
Degree $2$
Conductor $864$
Sign $0.292 - 0.956i$
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  + (0.721 + 1.21i)2-s + (−0.958 + 1.75i)4-s + (0.00931 − 0.0707i)5-s + (4.24 + 1.13i)7-s + (−2.82 + 0.100i)8-s + (0.0927 − 0.0397i)10-s + (3.52 − 4.59i)11-s + (4.11 − 3.15i)13-s + (1.68 + 5.98i)14-s + (−2.16 − 3.36i)16-s + 3.33i·17-s + (−0.207 − 0.0859i)19-s + (0.115 + 0.0841i)20-s + (8.14 + 0.973i)22-s + (−3.50 + 0.938i)23-s + ⋯
L(s)  = 1  + (0.510 + 0.860i)2-s + (−0.479 + 0.877i)4-s + (0.00416 − 0.0316i)5-s + (1.60 + 0.430i)7-s + (−0.999 + 0.0355i)8-s + (0.0293 − 0.0125i)10-s + (1.06 − 1.38i)11-s + (1.14 − 0.876i)13-s + (0.449 + 1.59i)14-s + (−0.540 − 0.841i)16-s + 0.808i·17-s + (−0.0476 − 0.0197i)19-s + (0.0257 + 0.0188i)20-s + (1.73 + 0.207i)22-s + (−0.730 + 0.195i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.292 - 0.956i$
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: \((2,\ 864,\ (\ :1/2),\ 0.292 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94047 + 1.43527i\)
\(L(\frac12)\) \(\approx\) \(1.94047 + 1.43527i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.721 - 1.21i)T \)
3 \( 1 \)
good5 \( 1 + (-0.00931 + 0.0707i)T + (-4.82 - 1.29i)T^{2} \)
7 \( 1 + (-4.24 - 1.13i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.52 + 4.59i)T + (-2.84 - 10.6i)T^{2} \)
13 \( 1 + (-4.11 + 3.15i)T + (3.36 - 12.5i)T^{2} \)
17 \( 1 - 3.33iT - 17T^{2} \)
19 \( 1 + (0.207 + 0.0859i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.50 - 0.938i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.74 - 0.492i)T + (28.0 - 7.50i)T^{2} \)
31 \( 1 + (1.34 - 2.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.86 - 2.01i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (6.48 - 1.73i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.23 - 1.60i)T + (-11.1 - 41.5i)T^{2} \)
47 \( 1 + (-2.50 + 1.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.65 + 4.00i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (0.511 - 3.88i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (-9.69 + 1.27i)T + (58.9 - 15.7i)T^{2} \)
67 \( 1 + (-4.45 - 5.80i)T + (-17.3 + 64.7i)T^{2} \)
71 \( 1 + (4.90 - 4.90i)T - 71iT^{2} \)
73 \( 1 + (11.0 + 11.0i)T + 73iT^{2} \)
79 \( 1 + (10.2 - 5.93i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.04 - 7.95i)T + (-80.1 + 21.4i)T^{2} \)
89 \( 1 + (-10.0 + 10.0i)T - 89iT^{2} \)
97 \( 1 + (3.80 + 6.59i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.51364135742116346240969979834, −8.903490365332225765479655142264, −8.509098078818719006017521703283, −7.990598582685162997380025953344, −6.75718222200439743686475479705, −5.82048994899810785070282323568, −5.30424748921234643349711998767, −4.08497118977378476957605261826, −3.27416883864063038945236003833, −1.42036661030461694960475642445, 1.36937263679915224705368674189, 2.07076801936430357423082068150, 3.83242921043438728902235006277, 4.42017521625701730966697388743, 5.21181980202214699733637531095, 6.48835496397259306861225492725, 7.34336904306153337054762781932, 8.584874172114556088617917860545, 9.224929264559438415979919118190, 10.20504504909382562908666885722

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