Properties

Label 2-2e7-4.3-c4-0-10
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.95i·3-s + 46.0·5-s − 57.5i·7-s + 17.7·9-s − 181. i·11-s + 1.33·13-s + 366. i·15-s − 62.6·17-s − 340. i·19-s + 457.·21-s + 447. i·23-s + 1.49e3·25-s + 785. i·27-s + 497.·29-s + 444. i·31-s + ⋯
L(s)  = 1  + 0.883i·3-s + 1.84·5-s − 1.17i·7-s + 0.219·9-s − 1.50i·11-s + 0.00787·13-s + 1.62i·15-s − 0.216·17-s − 0.942i·19-s + 1.03·21-s + 0.845i·23-s + 2.39·25-s + 1.07i·27-s + 0.591·29-s + 0.462i·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

\(L(\frac{5}{2})\) \(\approx\) \(2.49746\)
\(L(\frac12)\) \(\approx\) \(2.49746\)
\(L(3)\) 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 \( 1 - 7.95iT - 81T^{2} \)
5 \( 1 - 46.0T + 625T^{2} \)
7 \( 1 + 57.5iT - 2.40e3T^{2} \)
11 \( 1 + 181. iT - 1.46e4T^{2} \)
13 \( 1 - 1.33T + 2.85e4T^{2} \)
17 \( 1 + 62.6T + 8.35e4T^{2} \)
19 \( 1 + 340. iT - 1.30e5T^{2} \)
23 \( 1 - 447. iT - 2.79e5T^{2} \)
29 \( 1 - 497.T + 7.07e5T^{2} \)
31 \( 1 - 444. iT - 9.23e5T^{2} \)
37 \( 1 - 687.T + 1.87e6T^{2} \)
41 \( 1 + 3.07e3T + 2.82e6T^{2} \)
43 \( 1 + 307. iT - 3.41e6T^{2} \)
47 \( 1 - 4.04e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.97e3T + 7.89e6T^{2} \)
59 \( 1 - 2.87e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.53e3T + 1.38e7T^{2} \)
67 \( 1 - 3.96e3iT - 2.01e7T^{2} \)
71 \( 1 + 910. iT - 2.54e7T^{2} \)
73 \( 1 + 4.40e3T + 2.83e7T^{2} \)
79 \( 1 - 3.39e3iT - 3.89e7T^{2} \)
83 \( 1 + 8.38e3iT - 4.74e7T^{2} \)
89 \( 1 + 6.52e3T + 6.27e7T^{2} \)
97 \( 1 + 9.68e3T + 8.85e7T^{2} \)
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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

−13.12035043386775022655831830187, −11.14502599594813820249881558735, −10.42079219693108889182014882536, −9.691560594842368670427875260292, −8.754434884794595923434758532808, −6.96411467191121277970405932783, −5.83024714923566257182685804868, −4.67475350686703071827348980254, −3.14566764092306428004828900175, −1.21639678299000302856236050950, 1.71215176399189424347356942980, 2.32235744456316149041552490980, 4.96546062030563007489332825267, 6.11187036693194339327524249748, 6.90021636030635685963979784536, 8.437280446337638992428043228324, 9.635225003735827487947408124152, 10.21837140732050785949018540674, 12.06547157172008325233155750629, 12.67742719999470255387599723452

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