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Properties

Label 2-864-27.4-c1-0-20
Degree $2$
Conductor $864$
Sign $0.992 - 0.125i$
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  + (1.67 − 0.437i)3-s + (−0.158 + 0.899i)5-s + (1.25 − 1.04i)7-s + (2.61 − 1.46i)9-s + (0.887 + 5.03i)11-s + (1.45 − 0.531i)13-s + (0.127 + 1.57i)15-s + (−0.661 + 1.14i)17-s + (1.16 + 2.01i)19-s + (1.63 − 2.30i)21-s + (−2.86 − 2.40i)23-s + (3.91 + 1.42i)25-s + (3.74 − 3.60i)27-s + (0.0747 + 0.0272i)29-s + (−4.09 − 3.43i)31-s + ⋯
L(s)  = 1  + (0.967 − 0.252i)3-s + (−0.0709 + 0.402i)5-s + (0.472 − 0.396i)7-s + (0.872 − 0.488i)9-s + (0.267 + 1.51i)11-s + (0.404 − 0.147i)13-s + (0.0328 + 0.407i)15-s + (−0.160 + 0.277i)17-s + (0.266 + 0.461i)19-s + (0.357 − 0.503i)21-s + (−0.597 − 0.501i)23-s + (0.782 + 0.284i)25-s + (0.721 − 0.692i)27-s + (0.0138 + 0.00505i)29-s + (−0.735 − 0.617i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.992 - 0.125i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.992 - 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36649 + 0.148702i\)
\(L(\frac12)\) \(\approx\) \(2.36649 + 0.148702i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 + 0.437i)T \)
good5 \( 1 + (0.158 - 0.899i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-1.25 + 1.04i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.887 - 5.03i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-1.45 + 0.531i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.661 - 1.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.16 - 2.01i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.86 + 2.40i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-0.0747 - 0.0272i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.09 + 3.43i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-2.08 + 3.61i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.69 + 2.80i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.29 + 7.34i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (4.78 - 4.01i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + (1.03 - 5.89i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (1.68 - 1.41i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.0698 + 0.0254i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (3.73 - 6.47i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.83 + 6.64i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (14.7 + 5.36i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-1.07 - 0.392i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (9.09 + 15.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.75 - 9.93i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.13209274443519716275976472136, −9.296655015123253749843006624695, −8.476375002602573538296449355284, −7.44940762362022558042421812193, −7.17176926853000814823669997474, −5.95896159068825798286457835039, −4.49205739644477655769578250606, −3.84656625248509472039635520740, −2.53486842942441589424947380837, −1.51232000033677858754827251889, 1.29017181871373534954340400977, 2.72118003646833238919987869613, 3.64831644385087506943039352195, 4.70042793495965279168437574145, 5.66546717275054013659162206773, 6.79347668558420194617795379532, 7.934159293827483837696556760272, 8.566919409846642937941858403065, 9.053110621831487834952094527749, 9.972999011354961182147751649633

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