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Properties

Label 2-3e4-3.2-c4-0-4
Degree $2$
Conductor $81$
Sign $-i$
Analytic cond. $8.37296$
Root an. cond. $2.89360$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.355i·2-s + 15.8·4-s + 34.7i·5-s − 31.2·7-s + 11.3i·8-s − 12.3·10-s + 57.7i·11-s − 73.2·13-s − 11.0i·14-s + 249.·16-s + 386. i·17-s + 115.·19-s + 551. i·20-s − 20.5·22-s + 548. i·23-s + ⋯
L(s)  = 1  + 0.0887i·2-s + 0.992·4-s + 1.38i·5-s − 0.636·7-s + 0.176i·8-s − 0.123·10-s + 0.477i·11-s − 0.433·13-s − 0.0565i·14-s + 0.976·16-s + 1.33i·17-s + 0.320·19-s + 1.37i·20-s − 0.0423·22-s + 1.03i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-i$
Analytic conductor: \(8.37296\)
Root analytic conductor: \(2.89360\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :2),\ -i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.26542 + 1.26542i\)
\(L(\frac12)\) \(\approx\) \(1.26542 + 1.26542i\)
\(L(3)\) 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 \( 1 - 0.355iT - 16T^{2} \)
5 \( 1 - 34.7iT - 625T^{2} \)
7 \( 1 + 31.2T + 2.40e3T^{2} \)
11 \( 1 - 57.7iT - 1.46e4T^{2} \)
13 \( 1 + 73.2T + 2.85e4T^{2} \)
17 \( 1 - 386. iT - 8.35e4T^{2} \)
19 \( 1 - 115.T + 1.30e5T^{2} \)
23 \( 1 - 548. iT - 2.79e5T^{2} \)
29 \( 1 + 785. iT - 7.07e5T^{2} \)
31 \( 1 - 544.T + 9.23e5T^{2} \)
37 \( 1 - 898.T + 1.87e6T^{2} \)
41 \( 1 + 2.58e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.00e3T + 3.41e6T^{2} \)
47 \( 1 + 811. iT - 4.87e6T^{2} \)
53 \( 1 + 2.22e3iT - 7.89e6T^{2} \)
59 \( 1 + 1.51e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.90e3T + 1.38e7T^{2} \)
67 \( 1 - 4.50e3T + 2.01e7T^{2} \)
71 \( 1 + 3.99e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.43e3T + 2.83e7T^{2} \)
79 \( 1 - 1.20e3T + 3.89e7T^{2} \)
83 \( 1 - 9.25e3iT - 4.74e7T^{2} \)
89 \( 1 - 8.92e3iT - 6.27e7T^{2} \)
97 \( 1 - 6.67e3T + 8.85e7T^{2} \)
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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

−14.11938950532778707186227444525, −12.68992089878342869751826292514, −11.57998235919134988596878674906, −10.60625728739317738318378402139, −9.784809409408063665323478463209, −7.77826865202320006092025615217, −6.85518625228457858012774093009, −5.94799913548585162747859723215, −3.52728455625387193470344408134, −2.24761194384899778885424542625, 0.897709996041028160588406124645, 2.85692070950817403172740062021, 4.80896385816320441334366807889, 6.19437257383548092607266841211, 7.54193795528383286777337998416, 8.881693254616705948388824308920, 9.950004897289816815490846391040, 11.35447264678922011952162328127, 12.30472356172680321495311449562, 13.06142673105506396864433640125

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