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Properties

Label 2-864-288.11-c1-0-15
Degree $2$
Conductor $864$
Sign $0.839 + 0.543i$
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  + (0.484 − 1.32i)2-s + (−1.52 − 1.28i)4-s + (0.0345 + 0.262i)5-s + (0.779 + 2.90i)7-s + (−2.45 + 1.40i)8-s + (0.365 + 0.0814i)10-s + (0.904 − 0.694i)11-s + (−2.05 + 2.67i)13-s + (4.24 + 0.374i)14-s + (0.680 + 3.94i)16-s + 6.62·17-s + (5.15 − 2.13i)19-s + (0.285 − 0.446i)20-s + (−0.483 − 1.53i)22-s + (1.36 + 0.367i)23-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (−0.764 − 0.644i)4-s + (0.0154 + 0.117i)5-s + (0.294 + 1.09i)7-s + (−0.867 + 0.497i)8-s + (0.115 + 0.0257i)10-s + (0.272 − 0.209i)11-s + (−0.569 + 0.742i)13-s + (1.13 + 0.100i)14-s + (0.170 + 0.985i)16-s + 1.60·17-s + (1.18 − 0.490i)19-s + (0.0638 − 0.0998i)20-s + (−0.103 − 0.327i)22-s + (0.285 + 0.0765i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.839 + 0.543i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.839 + 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71564 - 0.507342i\)
\(L(\frac12)\) \(\approx\) \(1.71564 - 0.507342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.484 + 1.32i)T \)
3 \( 1 \)
good5 \( 1 + (-0.0345 - 0.262i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (-0.779 - 2.90i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.904 + 0.694i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (2.05 - 2.67i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 - 6.62T + 17T^{2} \)
19 \( 1 + (-5.15 + 2.13i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.36 - 0.367i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (5.03 + 0.662i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (-2.63 + 1.52i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.21 - 5.35i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.95 - 7.28i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.35 - 5.67i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (2.30 + 1.33i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.56 + 11.0i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (5.08 - 0.669i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (1.61 - 12.2i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (-4.48 + 5.84i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (3.28 + 3.28i)T + 71iT^{2} \)
73 \( 1 + (-2.26 + 2.26i)T - 73iT^{2} \)
79 \( 1 + (-7.66 + 13.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.17 + 0.154i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (7.53 - 7.53i)T - 89iT^{2} \)
97 \( 1 + (0.177 - 0.308i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.00732329117464309360584489418, −9.446144107934175769805912909582, −8.694641705185511970742934569760, −7.67302554726203954028618185548, −6.38861004972420941902492439053, −5.40304256216678141838635703811, −4.79454776747663198139823958651, −3.41652157140769060922328595677, −2.59743306781589252631621750239, −1.30012247943774956946641092958, 0.964267902440451900277655889315, 3.17377304898963958384868682329, 3.99972505256223996622345564952, 5.15020272428000429971053858335, 5.68766990567503808103375726251, 7.17387773241043548895938864023, 7.38714032469030542629151631138, 8.268100896189174814044151266937, 9.366924716379626426228637308658, 10.07271500671342789747584987487

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