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Properties

Label 2-864-72.61-c1-0-9
Degree $2$
Conductor $864$
Sign $-0.819 + 0.573i$
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  + (−1.73 + i)5-s + (2 − 3.46i)7-s + (−2.59 − 1.5i)11-s + (−1.73 + i)13-s − 5·17-s + i·19-s + (−1 − 1.73i)23-s + (−0.500 + 0.866i)25-s + (−2 − 3.46i)31-s + 7.99i·35-s − 2i·37-s + (−2.5 − 4.33i)41-s + (−9.52 − 5.5i)43-s + (3 − 5.19i)47-s + (−4.49 − 7.79i)49-s + ⋯
L(s)  = 1  + (−0.774 + 0.447i)5-s + (0.755 − 1.30i)7-s + (−0.783 − 0.452i)11-s + (−0.480 + 0.277i)13-s − 1.21·17-s + 0.229i·19-s + (−0.208 − 0.361i)23-s + (−0.100 + 0.173i)25-s + (−0.359 − 0.622i)31-s + 1.35i·35-s − 0.328i·37-s + (−0.390 − 0.676i)41-s + (−1.45 − 0.838i)43-s + (0.437 − 0.757i)47-s + (−0.642 − 1.11i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.819 + 0.573i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.819 + 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.157662 - 0.500042i\)
\(L(\frac12)\) \(\approx\) \(0.157662 - 0.500042i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1.73 - i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.73 - i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.52 + 5.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.3 + 6i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.59 + 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.46 + 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.08317514122404453840443298848, −8.833584202495667575159520870181, −7.943672545502807836665462349721, −7.37493368690980359687008298023, −6.61926547331536346519669681875, −5.22726069957269576993082820389, −4.30816242264194196807361304608, −3.52712294833341880930358252543, −2.07413641264551645508721876588, −0.23895999206140998992026907043, 1.90602253985529073899267565095, 2.96223421469609487127197059396, 4.53737965289165700850239042045, 4.99567968201802920328768033213, 6.05875121536404601880660099916, 7.28066183256177237714226907743, 8.124819463127232756879618203103, 8.647677066820465613742034573316, 9.531910307108321710998153146299, 10.58595342408425590333044320740

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