↑

Properties

Label 2-2e7-16.13-c11-0-27
Degree $2$
Conductor $128$
Sign $0.814 + 0.579i$
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  + (179. + 179. i)3-s + (−2.96e3 + 2.96e3i)5-s + 5.17e3i·7-s − 1.12e5i·9-s + (−6.94e5 + 6.94e5i)11-s + (−5.24e5 − 5.24e5i)13-s − 1.06e6·15-s + 1.82e6·17-s + (6.01e5 + 6.01e5i)19-s + (−9.30e5 + 9.30e5i)21-s − 2.32e7i·23-s + 3.12e7i·25-s + (5.20e7 − 5.20e7i)27-s + (−1.89e5 − 1.89e5i)29-s − 2.48e8·31-s + ⋯
L(s)  = 1  + (0.426 + 0.426i)3-s + (−0.423 + 0.423i)5-s + 0.116i·7-s − 0.635i·9-s + (−1.30 + 1.30i)11-s + (−0.391 − 0.391i)13-s − 0.361·15-s + 0.311·17-s + (0.0557 + 0.0557i)19-s + (−0.0497 + 0.0497i)21-s − 0.754i·23-s + 0.640i·25-s + (0.698 − 0.698i)27-s + (−0.00171 − 0.00171i)29-s − 1.56·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.814 + 0.579i$
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: \((2,\ 128,\ (\ :11/2),\ 0.814 + 0.579i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.358756173\)
\(L(\frac12)\) \(\approx\) \(1.358756173\)
\(L(\frac{13}{2})\) 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 \( 1 + (-179. - 179. i)T + 1.77e5iT^{2} \)
5 \( 1 + (2.96e3 - 2.96e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 5.17e3iT - 1.97e9T^{2} \)
11 \( 1 + (6.94e5 - 6.94e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (5.24e5 + 5.24e5i)T + 1.79e12iT^{2} \)
17 \( 1 - 1.82e6T + 3.42e13T^{2} \)
19 \( 1 + (-6.01e5 - 6.01e5i)T + 1.16e14iT^{2} \)
23 \( 1 + 2.32e7iT - 9.52e14T^{2} \)
29 \( 1 + (1.89e5 + 1.89e5i)T + 1.22e16iT^{2} \)
31 \( 1 + 2.48e8T + 2.54e16T^{2} \)
37 \( 1 + (-4.30e8 + 4.30e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 1.02e8iT - 5.50e17T^{2} \)
43 \( 1 + (-6.91e7 + 6.91e7i)T - 9.29e17iT^{2} \)
47 \( 1 - 2.50e8T + 2.47e18T^{2} \)
53 \( 1 + (-1.76e9 + 1.76e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (3.82e9 - 3.82e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (7.80e9 + 7.80e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (-6.53e9 - 6.53e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 1.53e10iT - 2.31e20T^{2} \)
73 \( 1 - 2.49e10iT - 3.13e20T^{2} \)
79 \( 1 - 2.46e10T + 7.47e20T^{2} \)
83 \( 1 + (-2.00e10 - 2.00e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 3.72e10iT - 2.77e21T^{2} \)
97 \( 1 + 1.06e11T + 7.15e21T^{2} \)
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

−10.93395110591150004799587900790, −10.05173902084298403431542872187, −9.170606388151863219726735601665, −7.84440316524253186389094475717, −7.09332120757131012504435452330, −5.54577534659847969881969109898, −4.35421793704811791969243539615, −3.20526218317707893550268995040, −2.18866865369537709238828503475, −0.35313748080223137008630606892, 0.821572692199953317786737144124, 2.22303174569975792304773947565, 3.30820355812715657546837592434, 4.77238698074622386061441596742, 5.81340048886123417218023310885, 7.43990649335049312482398089269, 8.035173336296252912824326378745, 8.996846357300210115318984230497, 10.41037231780757674296147224677, 11.28793349833608818457854167438

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