Properties

Label 2-864-72.11-c1-0-7
Degree $2$
Conductor $864$
Sign $0.441 + 0.897i$
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  + (0.565 + 0.978i)5-s + (−3.71 − 2.14i)7-s + (1.00 + 0.582i)11-s + (2.64 − 1.52i)13-s − 1.49i·17-s + 3.42·19-s + (−3.85 − 6.68i)23-s + (1.86 − 3.22i)25-s + (0.709 − 1.22i)29-s + (4.66 − 2.69i)31-s − 4.84i·35-s − 2.97i·37-s + (4.23 − 2.44i)41-s + (1.74 − 3.01i)43-s + (−1.77 + 3.08i)47-s + ⋯
L(s)  = 1  + (0.252 + 0.437i)5-s + (−1.40 − 0.810i)7-s + (0.304 + 0.175i)11-s + (0.733 − 0.423i)13-s − 0.362i·17-s + 0.785·19-s + (−0.804 − 1.39i)23-s + (0.372 − 0.644i)25-s + (0.131 − 0.228i)29-s + (0.837 − 0.483i)31-s − 0.819i·35-s − 0.488i·37-s + (0.661 − 0.381i)41-s + (0.265 − 0.460i)43-s + (−0.259 + 0.449i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.441 + 0.897i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.441 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09128 - 0.679183i\)
\(L(\frac12)\) \(\approx\) \(1.09128 - 0.679183i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.565 - 0.978i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.71 + 2.14i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.00 - 0.582i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.64 + 1.52i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.49iT - 17T^{2} \)
19 \( 1 - 3.42T + 19T^{2} \)
23 \( 1 + (3.85 + 6.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.709 + 1.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.66 + 2.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.97iT - 37T^{2} \)
41 \( 1 + (-4.23 + 2.44i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.74 + 3.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.77 - 3.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + (-7.50 + 4.33i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.16 + 1.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.58 - 9.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.54T + 71T^{2} \)
73 \( 1 + 7.06T + 73T^{2} \)
79 \( 1 + (-2.24 - 1.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.98 + 2.30i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.63iT - 89T^{2} \)
97 \( 1 + (-3.35 + 5.81i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.07993474760824511440999523637, −9.389020159457815646698700554285, −8.321080436621838151683983134135, −7.29985441555685171401466619185, −6.49402092585909944256212709906, −5.96342774466548734388054680414, −4.46260031546184812037712344365, −3.52156729761098020417963225444, −2.58361366427063090311032775481, −0.66995764621048348264482777424, 1.42256470701276481274939753294, 2.96074921698138298815554395451, 3.78291158537856948550184766188, 5.18413098612110981342046650999, 6.04176238074487916312449381591, 6.62977228105869773339911850629, 7.85522315719532557167639211720, 8.882899649259346655214099451736, 9.407369038532345528920324002771, 10.04639285940001059266314883186

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