↑

Properties

Label 2-2e7-16.5-c11-0-15
Degree $2$
Conductor $128$
Sign $-0.910 - 0.412i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−585. + 585. i)3-s + (6.85e3 + 6.85e3i)5-s + 1.99e4i·7-s − 5.08e5i·9-s + (9.64e4 + 9.64e4i)11-s + (6.74e5 − 6.74e5i)13-s − 8.03e6·15-s + 3.70e6·17-s + (2.61e6 − 2.61e6i)19-s + (−1.16e7 − 1.16e7i)21-s + 2.97e7i·23-s + 4.52e7i·25-s + (1.94e8 + 1.94e8i)27-s + (−4.14e7 + 4.14e7i)29-s + 6.07e7·31-s + ⋯
L(s)  = 1  + (−1.39 + 1.39i)3-s + (0.981 + 0.981i)5-s + 0.447i·7-s − 2.87i·9-s + (0.180 + 0.180i)11-s + (0.503 − 0.503i)13-s − 2.73·15-s + 0.633·17-s + (0.242 − 0.242i)19-s + (−0.623 − 0.623i)21-s + 0.964i·23-s + 0.927i·25-s + (2.60 + 2.60i)27-s + (−0.375 + 0.375i)29-s + 0.381·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.910 - 0.412i$
Analytic conductor: \(98.3479\)
Root analytic conductor: \(9.91705\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :11/2),\ -0.910 - 0.412i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.687215740\)
\(L(\frac12)\) \(\approx\) \(1.687215740\)
\(L(\frac{13}{2})\) 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 \( 1 + (585. - 585. i)T - 1.77e5iT^{2} \)
5 \( 1 + (-6.85e3 - 6.85e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 1.99e4iT - 1.97e9T^{2} \)
11 \( 1 + (-9.64e4 - 9.64e4i)T + 2.85e11iT^{2} \)
13 \( 1 + (-6.74e5 + 6.74e5i)T - 1.79e12iT^{2} \)
17 \( 1 - 3.70e6T + 3.42e13T^{2} \)
19 \( 1 + (-2.61e6 + 2.61e6i)T - 1.16e14iT^{2} \)
23 \( 1 - 2.97e7iT - 9.52e14T^{2} \)
29 \( 1 + (4.14e7 - 4.14e7i)T - 1.22e16iT^{2} \)
31 \( 1 - 6.07e7T + 2.54e16T^{2} \)
37 \( 1 + (-4.37e8 - 4.37e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 1.20e9iT - 5.50e17T^{2} \)
43 \( 1 + (-1.45e8 - 1.45e8i)T + 9.29e17iT^{2} \)
47 \( 1 - 1.52e9T + 2.47e18T^{2} \)
53 \( 1 + (-2.52e9 - 2.52e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (-4.54e9 - 4.54e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (9.96e7 - 9.96e7i)T - 4.35e19iT^{2} \)
67 \( 1 + (7.32e9 - 7.32e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 1.31e9iT - 2.31e20T^{2} \)
73 \( 1 - 2.59e9iT - 3.13e20T^{2} \)
79 \( 1 + 4.08e9T + 7.47e20T^{2} \)
83 \( 1 + (3.03e10 - 3.03e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 5.68e10iT - 2.77e21T^{2} \)
97 \( 1 + 5.73e10T + 7.15e21T^{2} \)
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

−11.38850817924389972639371497584, −10.53275535788361348353975504810, −9.933041009139782489576006616834, −9.059835057381006868508130398910, −6.97950506816806471350912359726, −5.83063119594569922658624053155, −5.50154606132537816670434837076, −4.01453862961912103841339525893, −2.86823506319501126803213074279, −1.01801101511384751242459217830, 0.58807331367023503528914982131, 1.18696431052996804930268412381, 2.15000345712639527047938658226, 4.53403182830965891636014054682, 5.64119268184588759353989264206, 6.23106879637948810416532862450, 7.35341540278160806085675796937, 8.482887330425348928037772066988, 9.909992481226220650152258975865, 11.01554689881563475701072663185

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