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Properties

Label 2-2e7-16.5-c3-0-6
Degree $2$
Conductor $128$
Sign $0.638 + 0.769i$
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  + (−1.98 + 1.98i)3-s + (0.596 + 0.596i)5-s − 29.0i·7-s + 19.1i·9-s + (−12.1 − 12.1i)11-s + (48.5 − 48.5i)13-s − 2.36·15-s + 86.7·17-s + (54.8 − 54.8i)19-s + (57.6 + 57.6i)21-s − 70.2i·23-s − 124. i·25-s + (−91.5 − 91.5i)27-s + (−63.4 + 63.4i)29-s − 8.86·31-s + ⋯
L(s)  = 1  + (−0.381 + 0.381i)3-s + (0.0533 + 0.0533i)5-s − 1.57i·7-s + 0.708i·9-s + (−0.332 − 0.332i)11-s + (1.03 − 1.03i)13-s − 0.0407·15-s + 1.23·17-s + (0.662 − 0.662i)19-s + (0.599 + 0.599i)21-s − 0.636i·23-s − 0.994i·25-s + (−0.652 − 0.652i)27-s + (−0.405 + 0.405i)29-s − 0.0513·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.638 + 0.769i$
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: \((2,\ 128,\ (\ :3/2),\ 0.638 + 0.769i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.24196 - 0.582957i\)
\(L(\frac12)\) \(\approx\) \(1.24196 - 0.582957i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.98 - 1.98i)T - 27iT^{2} \)
5 \( 1 + (-0.596 - 0.596i)T + 125iT^{2} \)
7 \( 1 + 29.0iT - 343T^{2} \)
11 \( 1 + (12.1 + 12.1i)T + 1.33e3iT^{2} \)
13 \( 1 + (-48.5 + 48.5i)T - 2.19e3iT^{2} \)
17 \( 1 - 86.7T + 4.91e3T^{2} \)
19 \( 1 + (-54.8 + 54.8i)T - 6.85e3iT^{2} \)
23 \( 1 + 70.2iT - 1.21e4T^{2} \)
29 \( 1 + (63.4 - 63.4i)T - 2.43e4iT^{2} \)
31 \( 1 + 8.86T + 2.97e4T^{2} \)
37 \( 1 + (-21.7 - 21.7i)T + 5.06e4iT^{2} \)
41 \( 1 - 153. iT - 6.89e4T^{2} \)
43 \( 1 + (-120. - 120. i)T + 7.95e4iT^{2} \)
47 \( 1 + 99.9T + 1.03e5T^{2} \)
53 \( 1 + (389. + 389. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-324. - 324. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-0.339 + 0.339i)T - 2.26e5iT^{2} \)
67 \( 1 + (565. - 565. i)T - 3.00e5iT^{2} \)
71 \( 1 - 419. iT - 3.57e5T^{2} \)
73 \( 1 + 374. iT - 3.89e5T^{2} \)
79 \( 1 + 705.T + 4.93e5T^{2} \)
83 \( 1 + (-947. + 947. i)T - 5.71e5iT^{2} \)
89 \( 1 - 4.72iT - 7.04e5T^{2} \)
97 \( 1 - 379.T + 9.12e5T^{2} \)
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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

−12.96086197391128673934360566261, −11.38211059756733363610314572579, −10.56267247921392005003166461713, −10.03855152871476880609293215611, −8.253881294055200845686423923097, −7.39934304522911988446874292177, −5.90551259786964633658365518329, −4.67449368919540837648613117311, −3.30884499405779814199535878726, −0.819946888747390395046436795876, 1.61647876772398661406250790784, 3.46499754019326157142521029023, 5.44646988377389589216888637692, 6.15341639803052256486861274233, 7.57273182072849968341819519850, 8.927541869769421286361721622351, 9.646085921879446667642555930443, 11.31924260076898907218556480846, 12.01478720230618545434002946776, 12.72331328154061228361843813115

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