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Properties

Label 2-3e4-9.4-c7-0-26
Degree $2$
Conductor $81$
Sign $-0.939 - 0.342i$
Analytic cond. $25.3031$
Root an. cond. $5.03022$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (64 − 110. i)4-s + (−881.5 − 1.52e3i)7-s + (−6.30e3 + 1.09e4i)13-s + (−8.19e3 − 1.41e4i)16-s + 1.43e4·19-s + (3.90e4 + 6.76e4i)25-s − 2.25e5·28-s + (−8.94e4 + 1.54e5i)31-s − 6.15e5·37-s + (−5.17e5 − 8.96e5i)43-s + (−1.14e6 + 1.97e6i)49-s + (8.06e5 + 1.39e6i)52-s + (−7.68e5 − 1.33e6i)61-s − 2.09e6·64-s + (2.02e6 − 3.51e6i)67-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (−0.971 − 1.68i)7-s + (−0.795 + 1.37i)13-s + (−0.499 − 0.866i)16-s + 0.480·19-s + (0.5 + 0.866i)25-s − 1.94·28-s + (−0.539 + 0.934i)31-s − 1.99·37-s + (−0.992 − 1.71i)43-s + (−1.38 + 2.40i)49-s + (0.795 + 1.37i)52-s + (−0.433 − 0.750i)61-s − 0.999·64-s + (0.824 − 1.42i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(25.3031\)
Root analytic conductor: \(5.03022\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :7/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0976825 + 0.553985i\)
\(L(\frac12)\) \(\approx\) \(0.0976825 + 0.553985i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-64 + 110. i)T^{2} \)
5 \( 1 + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (881.5 + 1.52e3i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (6.30e3 - 1.09e4i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + 4.10e8T^{2} \)
19 \( 1 - 1.43e4T + 8.93e8T^{2} \)
23 \( 1 + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + (8.94e4 - 1.54e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + 6.15e5T + 9.49e10T^{2} \)
41 \( 1 + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (5.17e5 + 8.96e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + 1.17e12T^{2} \)
59 \( 1 + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (7.68e5 + 1.33e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-2.02e6 + 3.51e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 9.09e12T^{2} \)
73 \( 1 - 1.23e6T + 1.10e13T^{2} \)
79 \( 1 + (-2.12e6 - 3.67e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + 4.42e13T^{2} \)
97 \( 1 + (2.63e6 + 4.56e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−12.28297793347449692626001099366, −10.99544463407257033313709632357, −10.15919742384712090213539427459, −9.285611456906382501216314120083, −7.12847392268299697358099191621, −6.77679090591027005771582096318, −5.02386681361590739073254449294, −3.51989589663641092076620618782, −1.61552683602046791291982804596, −0.17302871399128428759465496997, 2.43152442024036846578739891675, 3.25103718282067086768544514242, 5.34867335346696535616141597916, 6.53824844798646355358145685476, 7.901588492854894173373305963443, 8.950244195216344250981714363624, 10.15621396212794201208617892235, 11.69778709026514979993280761315, 12.44347144753196451718539013977, 13.07857520438373923668323509377

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