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Properties

Label 2-3e4-9.5-c4-0-1
Degree $2$
Conductor $81$
Sign $-0.642 - 0.766i$
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  + (2.59 − 1.5i)2-s + (−3.5 + 6.06i)4-s + (−28.5 − 16.5i)5-s + (9.5 + 16.4i)7-s + 69i·8-s − 99·10-s + (−106. + 61.5i)11-s + (−151 + 261. i)13-s + (49.3 + 28.5i)14-s + (47.5 + 82.2i)16-s − 414i·17-s − 304·19-s + (200. − 115. i)20-s + (−184.5 + 319. i)22-s + (259. + 150i)23-s + ⋯
L(s)  = 1  + (0.649 − 0.375i)2-s + (−0.218 + 0.378i)4-s + (−1.14 − 0.660i)5-s + (0.193 + 0.335i)7-s + 1.07i·8-s − 0.989·10-s + (−0.880 + 0.508i)11-s + (−0.893 + 1.54i)13-s + (0.251 + 0.145i)14-s + (0.185 + 0.321i)16-s − 1.43i·17-s − 0.842·19-s + (0.500 − 0.288i)20-s + (−0.381 + 0.660i)22-s + (0.491 + 0.283i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(8.37296\)
Root analytic conductor: \(2.89360\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :2),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.271253 + 0.581706i\)
\(L(\frac12)\) \(\approx\) \(0.271253 + 0.581706i\)
\(L(3)\) 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 \( 1 + (-2.59 + 1.5i)T + (8 - 13.8i)T^{2} \)
5 \( 1 + (28.5 + 16.5i)T + (312.5 + 541. i)T^{2} \)
7 \( 1 + (-9.5 - 16.4i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (106. - 61.5i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (151 - 261. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 414iT - 8.35e4T^{2} \)
19 \( 1 + 304T + 1.30e5T^{2} \)
23 \( 1 + (-259. - 150i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-587. + 339i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (119.5 - 206. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 740T + 1.87e6T^{2} \)
41 \( 1 + (-197. - 114i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-491 - 850. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.87e3 - 1.08e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 1.59e3iT - 7.89e6T^{2} \)
59 \( 1 + (2.53e3 + 1.46e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-158 - 273. i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (2.31e3 - 4.00e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 1.81e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.03e3T + 2.83e7T^{2} \)
79 \( 1 + (-5.22e3 - 9.04e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (1.09e4 - 6.31e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 7.00e3iT - 6.27e7T^{2} \)
97 \( 1 + (-3.25e3 - 5.64e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−13.85048040308462225908154597680, −12.69369319289772989748276852196, −11.98748424926168077034202498044, −11.29659438605459856303537985085, −9.403675541903555486963959011721, −8.287083217550770834161356139035, −7.25073189644504571120553456566, −4.94200794977494878192124117472, −4.33624709937230634221049811947, −2.55407982440465036166271079026, 0.25279443193821078283614096466, 3.24706856739956631434799988273, 4.58442461125484109628156794585, 5.93090228531331700338039955874, 7.34322715366949988407670569199, 8.309842030103785920579262106975, 10.31476768775889013631848588425, 10.81216324834310919878586956358, 12.46939068425767981827209563524, 13.22425556356380605670660101973

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