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Properties

Label 2-864-288.11-c1-0-17
Degree $2$
Conductor $864$
Sign $0.867 + 0.497i$
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  + (−1.39 − 0.217i)2-s + (1.90 + 0.607i)4-s + (−0.279 − 2.12i)5-s + (0.262 + 0.980i)7-s + (−2.53 − 1.26i)8-s + (−0.0707 + 3.02i)10-s + (−3.41 + 2.61i)11-s + (−0.912 + 1.18i)13-s + (−0.154 − 1.42i)14-s + (3.26 + 2.31i)16-s + 5.39·17-s + (6.75 − 2.79i)19-s + (0.756 − 4.21i)20-s + (5.33 − 2.91i)22-s + (2.26 + 0.606i)23-s + ⋯
L(s)  = 1  + (−0.988 − 0.153i)2-s + (0.952 + 0.303i)4-s + (−0.125 − 0.949i)5-s + (0.0992 + 0.370i)7-s + (−0.894 − 0.446i)8-s + (−0.0223 + 0.957i)10-s + (−1.02 + 0.789i)11-s + (−0.252 + 0.329i)13-s + (−0.0411 − 0.381i)14-s + (0.815 + 0.578i)16-s + 1.30·17-s + (1.54 − 0.641i)19-s + (0.169 − 0.942i)20-s + (1.13 − 0.622i)22-s + (0.471 + 0.126i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.867 + 0.497i$
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: \((2,\ 864,\ (\ :1/2),\ 0.867 + 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934233 - 0.248868i\)
\(L(\frac12)\) \(\approx\) \(0.934233 - 0.248868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.217i)T \)
3 \( 1 \)
good5 \( 1 + (0.279 + 2.12i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (-0.262 - 0.980i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.41 - 2.61i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (0.912 - 1.18i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 - 5.39T + 17T^{2} \)
19 \( 1 + (-6.75 + 2.79i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.26 - 0.606i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.40 - 0.579i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (8.30 - 4.79i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.47 + 5.98i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.03 + 3.87i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.15 + 6.71i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (-7.46 - 4.31i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.62 + 3.91i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.91 + 0.383i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (-0.0439 + 0.333i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (-3.21 + 4.19i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (-2.98 - 2.98i)T + 71iT^{2} \)
73 \( 1 + (1.45 - 1.45i)T - 73iT^{2} \)
79 \( 1 + (-4.91 + 8.50i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.17 + 0.285i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (2.17 - 2.17i)T - 89iT^{2} \)
97 \( 1 + (-4.24 + 7.35i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.953238135071077710471178222987, −9.198368735870760703415719973543, −8.596859859243884048576252178653, −7.53928818915024460836865148788, −7.16771518587497751512680822752, −5.55589025023949073783928295061, −5.00903890840095610040136015295, −3.40838218066609567642184828657, −2.22227538419678352445715637990, −0.883183891951665605094188462397, 0.990239879800501207894271483942, 2.75263270570167254227040951414, 3.36556349424250800837830611543, 5.29000001508119095955612778808, 5.98931773656279724234627895899, 7.18487218766858948075017373185, 7.66121270266034519166451217356, 8.350861900810424238637952509708, 9.616403446599829027485410049095, 10.17817290753110756805134942249

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