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Properties

Label 2-864-9.4-c1-0-6
Degree $2$
Conductor $864$
Sign $0.964 - 0.263i$
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  + (−0.5 + 0.866i)5-s + (0.724 + 1.25i)7-s + (−1.72 − 2.98i)11-s + (1.94 − 3.37i)13-s + 4.89·17-s + 4·19-s + (−0.275 + 0.476i)23-s + (2 + 3.46i)25-s + (4.94 + 8.57i)29-s + (−3.72 + 6.45i)31-s − 1.44·35-s + 8.89·37-s + (−1.05 + 1.81i)41-s + (−6.17 − 10.6i)43-s + (4.17 + 7.22i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.273 + 0.474i)7-s + (−0.520 − 0.900i)11-s + (0.540 − 0.936i)13-s + 1.18·17-s + 0.917·19-s + (−0.0573 + 0.0994i)23-s + (0.400 + 0.692i)25-s + (0.919 + 1.59i)29-s + (−0.668 + 1.15i)31-s − 0.245·35-s + 1.46·37-s + (−0.164 + 0.284i)41-s + (−0.941 − 1.63i)43-s + (0.608 + 1.05i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.964 - 0.263i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.964 - 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59238 + 0.213380i\)
\(L(\frac12)\) \(\approx\) \(1.59238 + 0.213380i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.724 - 1.25i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.72 + 2.98i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.94 + 3.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (0.275 - 0.476i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.94 - 8.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.72 - 6.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.89T + 37T^{2} \)
41 \( 1 + (1.05 - 1.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.17 + 10.6i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.17 - 7.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.898T + 53T^{2} \)
59 \( 1 + (0.174 - 0.301i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.949 + 1.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.17 + 2.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 + (4.27 + 7.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.72 + 4.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 + (2.94 + 5.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.44431400643910293942471874775, −9.293579126144827804380743803287, −8.413213391004515577523099924282, −7.77507043364901163209206900165, −6.82042267847475819365736316869, −5.58053964785478330885405675896, −5.22013230170831696402658597186, −3.46617477014396643123344031772, −2.97016851367770817159556250004, −1.14610365025289695387711388540, 1.04896443765932484233135236254, 2.49098791384464004328482122989, 3.95522948358889876889702046405, 4.63894662121938793634312902872, 5.70611857019374349861001660381, 6.76311793678875472035625793373, 7.75121113675165399701373122169, 8.208028299209905705019065133751, 9.535227684303215013317882159597, 9.913414115893251832425305863230

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