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Properties

Label 2-864-12.11-c1-0-7
Degree $2$
Conductor $864$
Sign $0.707 - 0.707i$
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  + 3.86i·5-s − 3.73i·7-s + 1.03·11-s + 4.46·13-s − 1.79i·17-s + 1.73i·19-s + 8.76·23-s − 9.92·25-s + 7.72i·29-s + 7.46i·31-s + 14.4·35-s + 0.464·37-s + 7.72i·41-s + 0.535i·43-s − 4.62·47-s + ⋯
L(s)  = 1  + 1.72i·5-s − 1.41i·7-s + 0.312·11-s + 1.23·13-s − 0.434i·17-s + 0.397i·19-s + 1.82·23-s − 1.98·25-s + 1.43i·29-s + 1.34i·31-s + 2.43·35-s + 0.0762·37-s + 1.20i·41-s + 0.0817i·43-s − 0.674·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53517 + 0.635892i\)
\(L(\frac12)\) \(\approx\) \(1.53517 + 0.635892i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.86iT - 5T^{2} \)
7 \( 1 + 3.73iT - 7T^{2} \)
11 \( 1 - 1.03T + 11T^{2} \)
13 \( 1 - 4.46T + 13T^{2} \)
17 \( 1 + 1.79iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 - 8.76T + 23T^{2} \)
29 \( 1 - 7.72iT - 29T^{2} \)
31 \( 1 - 7.46iT - 31T^{2} \)
37 \( 1 - 0.464T + 37T^{2} \)
41 \( 1 - 7.72iT - 41T^{2} \)
43 \( 1 - 0.535iT - 43T^{2} \)
47 \( 1 + 4.62T + 47T^{2} \)
53 \( 1 + 3.58iT - 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 + 6.26iT - 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 3.92T + 73T^{2} \)
79 \( 1 + 4.80iT - 79T^{2} \)
83 \( 1 - 2.07T + 83T^{2} \)
89 \( 1 + 1.79iT - 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−10.55728816689156425951895147082, −9.640645522740528078214661273717, −8.524110013904700481597875296305, −7.40309690261777729724321755670, −6.88726138037797557767089191384, −6.29646274296097223890378186499, −4.85402669076396054981651398784, −3.53999601395654524133627296247, −3.15577992997686144201284237328, −1.33103589153024903513582192508, 0.980048321834102687259846860406, 2.26187932849491843806780726019, 3.79868689301203313491683595175, 4.83213080679420169570466870787, 5.61742082544220368250818349308, 6.30256788223904596478981220303, 7.81079528060395379065824916018, 8.742248268928210567971665108640, 8.906976557094013555621432993518, 9.724583626457191934131124952338

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