↑

Properties

Label 2-2e7-4.3-c4-0-7
Degree $2$
Conductor $128$
Sign $1$
Analytic cond. $13.2313$
Root an. cond. $3.63749$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6.29i·3-s − 22.7·5-s − 51.5i·7-s + 41.3·9-s − 50.8i·11-s + 160.·13-s − 143. i·15-s + 416.·17-s + 515. i·19-s + 324.·21-s − 15.8i·23-s − 107.·25-s + 770. i·27-s + 979.·29-s − 1.90e3i·31-s + ⋯
L(s)  = 1  + 0.699i·3-s − 0.909·5-s − 1.05i·7-s + 0.510·9-s − 0.419i·11-s + 0.949·13-s − 0.636i·15-s + 1.44·17-s + 1.42i·19-s + 0.735·21-s − 0.0299i·23-s − 0.172·25-s + 1.05i·27-s + 1.16·29-s − 1.98i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $1$
Analytic conductor: \(13.2313\)
Root analytic conductor: \(3.63749\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.65388\)
\(L(\frac12)\) \(\approx\) \(1.65388\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 6.29iT - 81T^{2} \)
5 \( 1 + 22.7T + 625T^{2} \)
7 \( 1 + 51.5iT - 2.40e3T^{2} \)
11 \( 1 + 50.8iT - 1.46e4T^{2} \)
13 \( 1 - 160.T + 2.85e4T^{2} \)
17 \( 1 - 416.T + 8.35e4T^{2} \)
19 \( 1 - 515. iT - 1.30e5T^{2} \)
23 \( 1 + 15.8iT - 2.79e5T^{2} \)
29 \( 1 - 979.T + 7.07e5T^{2} \)
31 \( 1 + 1.90e3iT - 9.23e5T^{2} \)
37 \( 1 - 657.T + 1.87e6T^{2} \)
41 \( 1 - 2.83e3T + 2.82e6T^{2} \)
43 \( 1 + 2.53e3iT - 3.41e6T^{2} \)
47 \( 1 + 940. iT - 4.87e6T^{2} \)
53 \( 1 + 280.T + 7.89e6T^{2} \)
59 \( 1 + 5.03e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.88e3T + 1.38e7T^{2} \)
67 \( 1 - 1.42e3iT - 2.01e7T^{2} \)
71 \( 1 - 8.92e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.47e3T + 2.83e7T^{2} \)
79 \( 1 - 6.83e3iT - 3.89e7T^{2} \)
83 \( 1 + 4.78e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.70e3T + 6.27e7T^{2} \)
97 \( 1 - 1.89e3T + 8.85e7T^{2} \)
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

−12.57221492477218559505639106129, −11.46376323851410107058540786140, −10.50152639679465699948334130278, −9.758904599572012501446088527679, −8.192538560317594278863843628171, −7.44321729619736949932690180068, −5.86199533236918466224701057937, −4.14929514275081297234023564525, −3.65194786614921951072403757045, −0.934187368127885406595401334354, 1.15734026345308052042423487561, 2.95296985088972041612251133922, 4.58581243699050082093068756403, 6.10457370648218264826938260087, 7.29935282066560790115906378602, 8.208374686166841807717838422529, 9.315937931170718746963502730152, 10.74460674214700692926362044090, 11.96895958899947309502757751406, 12.38927637974850183701190910845

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