↑

Properties

Label 2-3e4-9.7-c7-0-17
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.874 − 1.51i)2-s + (62.4 + 108. i)4-s + (180. + 313. i)5-s + (758. − 1.31e3i)7-s + 442.·8-s + 632.·10-s + (3.31e3 − 5.74e3i)11-s + (589. + 1.02e3i)13-s + (−1.32e3 − 2.29e3i)14-s + (−7.60e3 + 1.31e4i)16-s + 1.36e4·17-s − 6.60e3·19-s + (−2.25e4 + 3.91e4i)20-s + (−5.79e3 − 1.00e4i)22-s + (3.84e4 + 6.66e4i)23-s + ⋯
L(s)  = 1  + (0.0772 − 0.133i)2-s + (0.488 + 0.845i)4-s + (0.646 + 1.12i)5-s + (0.836 − 1.44i)7-s + 0.305·8-s + 0.199·10-s + (0.751 − 1.30i)11-s + (0.0744 + 0.128i)13-s + (−0.129 − 0.223i)14-s + (−0.464 + 0.804i)16-s + 0.671·17-s − 0.221·19-s + (−0.631 + 1.09i)20-s + (−0.116 − 0.201i)22-s + (0.659 + 1.14i)23-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: $\chi_{81} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 81,\ (\ :7/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(4)\) \(\approx\) \(3.00765 + 0.530331i\)
\(L(\frac12)\) \(\approx\) \(3.00765 + 0.530331i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-0.874 + 1.51i)T + (-64 - 110. i)T^{2} \)
5 \( 1 + (-180. - 313. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-758. + 1.31e3i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-3.31e3 + 5.74e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (-589. - 1.02e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 - 1.36e4T + 4.10e8T^{2} \)
19 \( 1 + 6.60e3T + 8.93e8T^{2} \)
23 \( 1 + (-3.84e4 - 6.66e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-8.98e4 + 1.55e5i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + (9.44e4 + 1.63e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + 7.09e4T + 9.49e10T^{2} \)
41 \( 1 + (-3.33e5 - 5.78e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (3.70e5 - 6.41e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + (1.20e5 - 2.09e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + 7.42e4T + 1.17e12T^{2} \)
59 \( 1 + (-4.73e5 - 8.19e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-1.29e5 + 2.23e5i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.54e6 - 2.68e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 2.31e6T + 9.09e12T^{2} \)
73 \( 1 + 4.25e6T + 1.10e13T^{2} \)
79 \( 1 + (7.20e5 - 1.24e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-1.70e6 + 2.94e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + 1.28e7T + 4.42e13T^{2} \)
97 \( 1 + (-7.34e6 + 1.27e7i)T + (-4.03e13 - 6.99e13i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.26828240493525326858802814619, −11.39430360273884592302836931814, −11.15467288066235005480676342759, −9.913875207953004859353148457652, −8.157411040578068480088530760847, −7.21249694146798162582878753735, −6.16297238985876176509885698756, −4.06148122684138018417797708858, −2.98233828971132038570148447758, −1.30205059357215751273072543186, 1.29822755631907597600567407190, 2.14011781811955285807540004045, 4.89834451444211948165240662779, 5.43998618687399707387365344750, 6.83301192653937651182055301912, 8.618346375317102734646583338203, 9.357447338340628044290901525822, 10.60063906344238528990195777504, 12.06361448992032586541587541385, 12.56450024060096422697922030358

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