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Properties

Label 2-2e7-8.5-c7-0-5
Degree $2$
Conductor $128$
Sign $1$
Analytic cond. $39.9852$
Root an. cond. $6.32339$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 82.2i·3-s + 232. i·5-s − 1.60e3·7-s − 4.57e3·9-s − 3.37e3i·11-s + 1.09e4i·13-s + 1.91e4·15-s + 2.31e4·17-s − 2.47e4i·19-s + 1.32e5i·21-s − 4.38e4·23-s + 2.40e4·25-s + 1.96e5i·27-s + 7.97e4i·29-s + 1.08e5·31-s + ⋯
L(s)  = 1  − 1.75i·3-s + 0.831i·5-s − 1.77·7-s − 2.09·9-s − 0.763i·11-s + 1.37i·13-s + 1.46·15-s + 1.14·17-s − 0.827i·19-s + 3.11i·21-s − 0.751·23-s + 0.307·25-s + 1.91i·27-s + 0.607i·29-s + 0.654·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $1$
Analytic conductor: \(39.9852\)
Root analytic conductor: \(6.32339\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.049025463\)
\(L(\frac12)\) \(\approx\) \(1.049025463\)
\(L(\frac{9}{2})\) 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 \( 1 + 82.2iT - 2.18e3T^{2} \)
5 \( 1 - 232. iT - 7.81e4T^{2} \)
7 \( 1 + 1.60e3T + 8.23e5T^{2} \)
11 \( 1 + 3.37e3iT - 1.94e7T^{2} \)
13 \( 1 - 1.09e4iT - 6.27e7T^{2} \)
17 \( 1 - 2.31e4T + 4.10e8T^{2} \)
19 \( 1 + 2.47e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.38e4T + 3.40e9T^{2} \)
29 \( 1 - 7.97e4iT - 1.72e10T^{2} \)
31 \( 1 - 1.08e5T + 2.75e10T^{2} \)
37 \( 1 + 1.73e5iT - 9.49e10T^{2} \)
41 \( 1 - 4.34e4T + 1.94e11T^{2} \)
43 \( 1 - 6.09e5iT - 2.71e11T^{2} \)
47 \( 1 + 3.18e4T + 5.06e11T^{2} \)
53 \( 1 + 1.98e6iT - 1.17e12T^{2} \)
59 \( 1 - 1.92e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.63e6iT - 3.14e12T^{2} \)
67 \( 1 - 1.97e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.01e6T + 9.09e12T^{2} \)
73 \( 1 - 2.39e6T + 1.10e13T^{2} \)
79 \( 1 + 2.37e6T + 1.92e13T^{2} \)
83 \( 1 - 2.95e6iT - 2.71e13T^{2} \)
89 \( 1 + 7.18e6T + 4.42e13T^{2} \)
97 \( 1 - 1.49e7T + 8.07e13T^{2} \)
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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

−12.16011015430612815887390014908, −11.26372255971652606153479209729, −9.852284619023557890555330231408, −8.658979442724825750217659810549, −7.26643069892315095393603121875, −6.64448266740675839889892121655, −5.96905331021797583736301986532, −3.37463821899618841132037612720, −2.46470447679173503291400570975, −0.863353193492562278944912926126, 0.40039045385511207751409877890, 3.02785745509486201113163611470, 3.88787100680286940941639866418, 5.14559617063660812147234781342, 6.06728520208443802084429180561, 8.005102284992406800525789284626, 9.236692046357756271924028647319, 9.989782798576779286203672621631, 10.35959133229660330484651330120, 12.14106588740784485504873790842

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