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Properties

Label 2-3e3-3.2-c14-0-18
Degree $2$
Conductor $27$
Sign $-i$
Analytic cond. $33.5688$
Root an. cond. $5.79386$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 185. i·2-s − 1.79e4·4-s − 1.15e5i·5-s − 3.69e5·7-s + 2.81e5i·8-s − 2.14e7·10-s − 3.69e7i·11-s + 6.11e7·13-s + 6.83e7i·14-s − 2.41e8·16-s − 2.72e8i·17-s − 6.92e8·19-s + 2.07e9i·20-s − 6.83e9·22-s + 1.35e9i·23-s + ⋯
L(s)  = 1  − 1.44i·2-s − 1.09·4-s − 1.47i·5-s − 0.448·7-s + 0.134i·8-s − 2.14·10-s − 1.89i·11-s + 0.975·13-s + 0.648i·14-s − 0.898·16-s − 0.664i·17-s − 0.775·19-s + 1.61i·20-s − 2.74·22-s + 0.398i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-i$
Analytic conductor: \(33.5688\)
Root analytic conductor: \(5.79386\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :7),\ -i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(1.446993413\)
\(L(\frac12)\) \(\approx\) \(1.446993413\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + 185. iT - 1.63e4T^{2} \)
5 \( 1 + 1.15e5iT - 6.10e9T^{2} \)
7 \( 1 + 3.69e5T + 6.78e11T^{2} \)
11 \( 1 + 3.69e7iT - 3.79e14T^{2} \)
13 \( 1 - 6.11e7T + 3.93e15T^{2} \)
17 \( 1 + 2.72e8iT - 1.68e17T^{2} \)
19 \( 1 + 6.92e8T + 7.99e17T^{2} \)
23 \( 1 - 1.35e9iT - 1.15e19T^{2} \)
29 \( 1 - 9.06e9iT - 2.97e20T^{2} \)
31 \( 1 - 1.67e10T + 7.56e20T^{2} \)
37 \( 1 - 1.75e11T + 9.01e21T^{2} \)
41 \( 1 - 1.20e11iT - 3.79e22T^{2} \)
43 \( 1 - 3.35e11T + 7.38e22T^{2} \)
47 \( 1 + 8.58e11iT - 2.56e23T^{2} \)
53 \( 1 - 2.24e12iT - 1.37e24T^{2} \)
59 \( 1 - 2.61e12iT - 6.19e24T^{2} \)
61 \( 1 - 1.86e12T + 9.87e24T^{2} \)
67 \( 1 + 5.44e12T + 3.67e25T^{2} \)
71 \( 1 + 6.80e12iT - 8.27e25T^{2} \)
73 \( 1 - 6.05e12T + 1.22e26T^{2} \)
79 \( 1 + 1.60e13T + 3.68e26T^{2} \)
83 \( 1 + 2.87e13iT - 7.36e26T^{2} \)
89 \( 1 + 3.16e13iT - 1.95e27T^{2} \)
97 \( 1 + 1.09e14T + 6.52e27T^{2} \)
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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

−13.03964459788278042298510036346, −11.82058208329477516416629898961, −10.83702595067159411812936563424, −9.296769104815267501901474974941, −8.486232685125330655220077425015, −5.92902487965472995038281541162, −4.26361997311085343510539433112, −3.00197724856828811151465210332, −1.20258580936126269035939113281, −0.48690579782897725405669055107, 2.35204006092108059810810320010, 4.24138880662367272586306045439, 6.17385784515120228842873929778, 6.83818535013213989447587070938, 7.981156649797787977101877673454, 9.733830033921109569387928374266, 11.03508994485358839197220136112, 12.89942971700540515365907856219, 14.40029830107221768700711672774, 15.02996484435617994416818835821

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