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Properties

Label 2-2e7-16.13-c11-0-36
Degree $2$
Conductor $128$
Sign $0.430 + 0.902i$
Analytic cond. $98.3479$
Root an. cond. $9.91705$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (145. + 145. i)3-s + (4.65e3 − 4.65e3i)5-s − 4.32e4i·7-s − 1.35e5i·9-s + (6.85e5 − 6.85e5i)11-s + (1.40e6 + 1.40e6i)13-s + 1.34e6·15-s + 3.84e6·17-s + (7.81e6 + 7.81e6i)19-s + (6.27e6 − 6.27e6i)21-s + 1.99e7i·23-s + 5.53e6i·25-s + (4.52e7 − 4.52e7i)27-s + (−1.18e8 − 1.18e8i)29-s + 5.32e7·31-s + ⋯
L(s)  = 1  + (0.344 + 0.344i)3-s + (0.665 − 0.665i)5-s − 0.972i·7-s − 0.762i·9-s + (1.28 − 1.28i)11-s + (1.04 + 1.04i)13-s + 0.459·15-s + 0.656·17-s + (0.724 + 0.724i)19-s + (0.335 − 0.335i)21-s + 0.647i·23-s + 0.113i·25-s + (0.607 − 0.607i)27-s + (−1.07 − 1.07i)29-s + 0.333·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.430 + 0.902i$
Analytic conductor: \(98.3479\)
Root analytic conductor: \(9.91705\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :11/2),\ 0.430 + 0.902i)\)

Particular Values

\(L(6)\) \(\approx\) \(3.639265380\)
\(L(\frac12)\) \(\approx\) \(3.639265380\)
\(L(\frac{13}{2})\) 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 \( 1 + (-145. - 145. i)T + 1.77e5iT^{2} \)
5 \( 1 + (-4.65e3 + 4.65e3i)T - 4.88e7iT^{2} \)
7 \( 1 + 4.32e4iT - 1.97e9T^{2} \)
11 \( 1 + (-6.85e5 + 6.85e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (-1.40e6 - 1.40e6i)T + 1.79e12iT^{2} \)
17 \( 1 - 3.84e6T + 3.42e13T^{2} \)
19 \( 1 + (-7.81e6 - 7.81e6i)T + 1.16e14iT^{2} \)
23 \( 1 - 1.99e7iT - 9.52e14T^{2} \)
29 \( 1 + (1.18e8 + 1.18e8i)T + 1.22e16iT^{2} \)
31 \( 1 - 5.32e7T + 2.54e16T^{2} \)
37 \( 1 + (-3.70e8 + 3.70e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 8.83e7iT - 5.50e17T^{2} \)
43 \( 1 + (1.81e8 - 1.81e8i)T - 9.29e17iT^{2} \)
47 \( 1 + 1.92e9T + 2.47e18T^{2} \)
53 \( 1 + (1.39e9 - 1.39e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-2.09e9 + 2.09e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (1.23e9 + 1.23e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (-9.73e9 - 9.73e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 2.30e9iT - 2.31e20T^{2} \)
73 \( 1 + 2.18e10iT - 3.13e20T^{2} \)
79 \( 1 + 1.21e8T + 7.47e20T^{2} \)
83 \( 1 + (-4.69e10 - 4.69e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 6.68e10iT - 2.77e21T^{2} \)
97 \( 1 - 4.52e10T + 7.15e21T^{2} \)
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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

−11.10734360057438254130274386506, −9.578987108108058213370031672211, −9.262249048909235902040179958883, −8.063160865785066326583248714350, −6.54716776508725742602682829960, −5.72243934399976361426846682626, −3.99194389186157136533100185400, −3.54823078440090591686256060825, −1.43677345117368902123573066606, −0.860870988394171880973586371237, 1.29174940854839947541360933462, 2.24830827726836225959054582787, 3.25813584469650531524040989966, 4.97062261193287918861374827326, 6.09881579818333828013253928229, 7.09293651081980503633343727523, 8.275441916174171889313832700894, 9.337072229617622124342018103970, 10.28635512940773307739494740832, 11.41377117602484574157248093887

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