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Properties

Label 2-2e7-16.5-c3-0-0
Degree $2$
Conductor $128$
Sign $-0.954 + 0.298i$
Analytic cond. $7.55224$
Root an. cond. $2.74813$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.49 + 5.49i)3-s + (4.66 + 4.66i)5-s + 24.8i·7-s − 33.4i·9-s + (−22.3 − 22.3i)11-s + (11.2 − 11.2i)13-s − 51.2·15-s − 88.4·17-s + (−37.8 + 37.8i)19-s + (−136. − 136. i)21-s − 48.1i·23-s − 81.4i·25-s + (35.2 + 35.2i)27-s + (−10.4 + 10.4i)29-s − 96.9·31-s + ⋯
L(s)  = 1  + (−1.05 + 1.05i)3-s + (0.417 + 0.417i)5-s + 1.34i·7-s − 1.23i·9-s + (−0.612 − 0.612i)11-s + (0.240 − 0.240i)13-s − 0.882·15-s − 1.26·17-s + (−0.456 + 0.456i)19-s + (−1.42 − 1.42i)21-s − 0.436i·23-s − 0.651i·25-s + (0.251 + 0.251i)27-s + (−0.0668 + 0.0668i)29-s − 0.561·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.954 + 0.298i$
Analytic conductor: \(7.55224\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.954 + 0.298i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0856353 - 0.560739i\)
\(L(\frac12)\) \(\approx\) \(0.0856353 - 0.560739i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.49 - 5.49i)T - 27iT^{2} \)
5 \( 1 + (-4.66 - 4.66i)T + 125iT^{2} \)
7 \( 1 - 24.8iT - 343T^{2} \)
11 \( 1 + (22.3 + 22.3i)T + 1.33e3iT^{2} \)
13 \( 1 + (-11.2 + 11.2i)T - 2.19e3iT^{2} \)
17 \( 1 + 88.4T + 4.91e3T^{2} \)
19 \( 1 + (37.8 - 37.8i)T - 6.85e3iT^{2} \)
23 \( 1 + 48.1iT - 1.21e4T^{2} \)
29 \( 1 + (10.4 - 10.4i)T - 2.43e4iT^{2} \)
31 \( 1 + 96.9T + 2.97e4T^{2} \)
37 \( 1 + (-163. - 163. i)T + 5.06e4iT^{2} \)
41 \( 1 - 360. iT - 6.89e4T^{2} \)
43 \( 1 + (-100. - 100. i)T + 7.95e4iT^{2} \)
47 \( 1 - 220.T + 1.03e5T^{2} \)
53 \( 1 + (-175. - 175. i)T + 1.48e5iT^{2} \)
59 \( 1 + (405. + 405. i)T + 2.05e5iT^{2} \)
61 \( 1 + (664. - 664. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-107. + 107. i)T - 3.00e5iT^{2} \)
71 \( 1 + 215. iT - 3.57e5T^{2} \)
73 \( 1 - 668. iT - 3.89e5T^{2} \)
79 \( 1 + 822.T + 4.93e5T^{2} \)
83 \( 1 + (326. - 326. i)T - 5.71e5iT^{2} \)
89 \( 1 - 262. iT - 7.04e5T^{2} \)
97 \( 1 + 150.T + 9.12e5T^{2} \)
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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

−13.30796016277823517049035231556, −12.17183781429452488638315921646, −11.15201964668679165336493586160, −10.53361968770122518577552848026, −9.433904380478434981445955642950, −8.366730138174878508931598474522, −6.29922655138275257944057870642, −5.69831681608103246984630230701, −4.49656496118428366749861012970, −2.64964379595915999850412282114, 0.32077704262912549182581296547, 1.79459591369877468177050766890, 4.36358873998188202951626107963, 5.65511563553257886594478863560, 6.88762214426423762148697854810, 7.50775291587233785255579373246, 9.150993341499012995895629922112, 10.59024398113841472604536685115, 11.21389116595962735938126809071, 12.51218206937968703235043140409

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