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Properties

Label 2-864-288.131-c1-0-15
Degree $2$
Conductor $864$
Sign $0.867 - 0.498i$
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  + (−1.27 + 0.614i)2-s + (1.24 − 1.56i)4-s + (−0.0452 + 0.343i)5-s + (0.358 − 1.33i)7-s + (−0.622 + 2.75i)8-s + (−0.153 − 0.465i)10-s + (1.53 + 1.17i)11-s + (1.07 + 1.39i)13-s + (0.365 + 1.92i)14-s + (−0.902 − 3.89i)16-s + 0.574·17-s + (−1.27 − 0.527i)19-s + (0.481 + 0.498i)20-s + (−2.68 − 0.557i)22-s + (−2.80 + 0.751i)23-s + ⋯
L(s)  = 1  + (−0.900 + 0.434i)2-s + (0.622 − 0.782i)4-s + (−0.0202 + 0.153i)5-s + (0.135 − 0.505i)7-s + (−0.220 + 0.975i)8-s + (−0.0485 − 0.147i)10-s + (0.463 + 0.355i)11-s + (0.297 + 0.387i)13-s + (0.0977 + 0.514i)14-s + (−0.225 − 0.974i)16-s + 0.139·17-s + (−0.292 − 0.120i)19-s + (0.107 + 0.111i)20-s + (−0.572 − 0.118i)22-s + (−0.584 + 0.156i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.867 - 0.498i$
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: \((2,\ 864,\ (\ :1/2),\ 0.867 - 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04052 + 0.277572i\)
\(L(\frac12)\) \(\approx\) \(1.04052 + 0.277572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.614i)T \)
3 \( 1 \)
good5 \( 1 + (0.0452 - 0.343i)T + (-4.82 - 1.29i)T^{2} \)
7 \( 1 + (-0.358 + 1.33i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.53 - 1.17i)T + (2.84 + 10.6i)T^{2} \)
13 \( 1 + (-1.07 - 1.39i)T + (-3.36 + 12.5i)T^{2} \)
17 \( 1 - 0.574T + 17T^{2} \)
19 \( 1 + (1.27 + 0.527i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.80 - 0.751i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.67 + 0.352i)T + (28.0 - 7.50i)T^{2} \)
31 \( 1 + (-4.87 - 2.81i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.837 - 2.02i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.97 + 11.0i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-4.52 + 5.89i)T + (-11.1 - 41.5i)T^{2} \)
47 \( 1 + (-4.21 + 2.43i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.53 - 8.52i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (5.62 + 0.740i)T + (56.9 + 15.2i)T^{2} \)
61 \( 1 + (-0.645 - 4.90i)T + (-58.9 + 15.7i)T^{2} \)
67 \( 1 + (-4.11 - 5.35i)T + (-17.3 + 64.7i)T^{2} \)
71 \( 1 + (-5.04 + 5.04i)T - 71iT^{2} \)
73 \( 1 + (-8.79 - 8.79i)T + 73iT^{2} \)
79 \( 1 + (6.57 + 11.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.3 + 1.49i)T + (80.1 - 21.4i)T^{2} \)
89 \( 1 + (-8.74 - 8.74i)T + 89iT^{2} \)
97 \( 1 + (-0.426 - 0.738i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.35943700708506797867404248717, −9.210519419440665713631693470024, −8.669781510851973931088488640825, −7.63191372653305651282475103367, −6.94250509110042033683790152281, −6.19954158706862788539019254764, −5.06675421055652741223042955125, −3.91518258543773829391425585093, −2.38100018614211371071563562716, −1.04737490000011959607724746732, 0.945160787490851249529534442870, 2.34562131920488048091543846517, 3.39856135658165135920051964016, 4.58756705304446084748842172183, 5.98842642450684161405126640207, 6.69445945614565302018680873565, 7.955124146155263882382740946105, 8.410364935981151050964239576225, 9.265134827912624568047325935816, 10.02842470892124541051938098265

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