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Properties

Label 2-864-288.131-c1-0-14
Degree $2$
Conductor $864$
Sign $0.0367 - 0.999i$
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  + (0.259 + 1.39i)2-s + (−1.86 + 0.720i)4-s + (0.173 − 1.31i)5-s + (0.0444 − 0.165i)7-s + (−1.48 − 2.40i)8-s + (1.87 − 0.100i)10-s + (0.877 + 0.673i)11-s + (2.54 + 3.32i)13-s + (0.242 + 0.0187i)14-s + (2.96 − 2.68i)16-s − 1.67·17-s + (3.90 + 1.61i)19-s + (0.626 + 2.58i)20-s + (−0.708 + 1.39i)22-s + (−1.65 + 0.443i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.983i)2-s + (−0.932 + 0.360i)4-s + (0.0776 − 0.590i)5-s + (0.0167 − 0.0626i)7-s + (−0.525 − 0.850i)8-s + (0.594 − 0.0317i)10-s + (0.264 + 0.203i)11-s + (0.706 + 0.921i)13-s + (0.0646 + 0.00502i)14-s + (0.740 − 0.672i)16-s − 0.406·17-s + (0.895 + 0.370i)19-s + (0.140 + 0.578i)20-s + (−0.151 + 0.297i)22-s + (−0.344 + 0.0924i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0367 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0367 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.0367 - 0.999i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.0367 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14968 + 1.10816i\)
\(L(\frac12)\) \(\approx\) \(1.14968 + 1.10816i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.259 - 1.39i)T \)
3 \( 1 \)
good5 \( 1 + (-0.173 + 1.31i)T + (-4.82 - 1.29i)T^{2} \)
7 \( 1 + (-0.0444 + 0.165i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.877 - 0.673i)T + (2.84 + 10.6i)T^{2} \)
13 \( 1 + (-2.54 - 3.32i)T + (-3.36 + 12.5i)T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
19 \( 1 + (-3.90 - 1.61i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.65 - 0.443i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-6.46 + 0.851i)T + (28.0 - 7.50i)T^{2} \)
31 \( 1 + (-3.09 - 1.78i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.58 - 8.64i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.790 - 2.94i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.37 - 7.00i)T + (-11.1 - 41.5i)T^{2} \)
47 \( 1 + (-0.598 + 0.345i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.43 + 10.7i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.24 + 0.427i)T + (56.9 + 15.2i)T^{2} \)
61 \( 1 + (1.28 + 9.76i)T + (-58.9 + 15.7i)T^{2} \)
67 \( 1 + (5.81 + 7.57i)T + (-17.3 + 64.7i)T^{2} \)
71 \( 1 + (0.107 - 0.107i)T - 71iT^{2} \)
73 \( 1 + (-3.93 - 3.93i)T + 73iT^{2} \)
79 \( 1 + (-0.777 - 1.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.82 - 0.635i)T + (80.1 - 21.4i)T^{2} \)
89 \( 1 + (-10.7 - 10.7i)T + 89iT^{2} \)
97 \( 1 + (6.90 + 11.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.01055370128800591259836384832, −9.358397163145315651832222072553, −8.532833164088728137712986965408, −7.911606715018152479875211548533, −6.72296149502825243482691049048, −6.24035351702115587744933030693, −4.99402019126309845043979103586, −4.41493164809294752550790207138, −3.23307807620446676479100831192, −1.25655251750395832094214404980, 0.912582160127559515095939494002, 2.49284205818778475553729606520, 3.30468741429956944135865867315, 4.34787759771707918017269177421, 5.47470098693924054846914561446, 6.28856149648994433383519356742, 7.48629326425394026697174255965, 8.561831186924186025566099638318, 9.190832345815988236118684113952, 10.36663160852125292254735912652

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