Properties

Degree $2$
Conductor $128$
Sign $0.269 - 0.962i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.86 − 7.86i)3-s + (−27.2 − 27.2i)5-s − 50.3·7-s + 42.8i·9-s + (−53.1 + 53.1i)11-s + (125. − 125. i)13-s + 428. i·15-s + 286.·17-s + (−99.5 − 99.5i)19-s + (395. + 395. i)21-s + 100.·23-s + 858. i·25-s + (−300. + 300. i)27-s + (−343. + 343. i)29-s − 208. i·31-s + ⋯
L(s)  = 1  + (−0.874 − 0.874i)3-s + (−1.08 − 1.08i)5-s − 1.02·7-s + 0.528i·9-s + (−0.438 + 0.438i)11-s + (0.740 − 0.740i)13-s + 1.90i·15-s + 0.990·17-s + (−0.275 − 0.275i)19-s + (0.897 + 0.897i)21-s + 0.189·23-s + 1.37i·25-s + (−0.412 + 0.412i)27-s + (−0.408 + 0.408i)29-s − 0.216i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.269 - 0.962i$
Motivic weight: \(4\)
Character: $\chi_{128} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ 0.269 - 0.962i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0282131 + 0.0213914i\)
\(L(\frac12)\) \(\approx\) \(0.0282131 + 0.0213914i\)
\(L(3)\) 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 \( 1 + (7.86 + 7.86i)T + 81iT^{2} \)
5 \( 1 + (27.2 + 27.2i)T + 625iT^{2} \)
7 \( 1 + 50.3T + 2.40e3T^{2} \)
11 \( 1 + (53.1 - 53.1i)T - 1.46e4iT^{2} \)
13 \( 1 + (-125. + 125. i)T - 2.85e4iT^{2} \)
17 \( 1 - 286.T + 8.35e4T^{2} \)
19 \( 1 + (99.5 + 99.5i)T + 1.30e5iT^{2} \)
23 \( 1 - 100.T + 2.79e5T^{2} \)
29 \( 1 + (343. - 343. i)T - 7.07e5iT^{2} \)
31 \( 1 + 208. iT - 9.23e5T^{2} \)
37 \( 1 + (-1.15e3 - 1.15e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 2.33e3iT - 2.82e6T^{2} \)
43 \( 1 + (2.07e3 - 2.07e3i)T - 3.41e6iT^{2} \)
47 \( 1 - 1.05e3iT - 4.87e6T^{2} \)
53 \( 1 + (2.13e3 + 2.13e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-3.72e3 + 3.72e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (2.49e3 - 2.49e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (329. + 329. i)T + 2.01e7iT^{2} \)
71 \( 1 - 1.04e3T + 2.54e7T^{2} \)
73 \( 1 - 2.67e3iT - 2.83e7T^{2} \)
79 \( 1 + 4.47e3iT - 3.89e7T^{2} \)
83 \( 1 + (-1.45e3 - 1.45e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 1.14e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.31e4T + 8.85e7T^{2} \)
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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

−12.77768535809740531596377093206, −12.07628568341549514369924282768, −11.11019623880602120293121764459, −9.720524605417795671963882228134, −8.361927933300942774314178748662, −7.42328177669469192246948661163, −6.22494828293903749446644224279, −5.07687111995607399155339821423, −3.49314363595689298207532235237, −1.02369355814731790407857049875, 0.02097019397243991078038220246, 3.19522180578392609807596019307, 4.11561722227325180689785665719, 5.75202728336064569654682061527, 6.75795987342416577540890226334, 8.025612188141757481594648054367, 9.614500985400230752192582120931, 10.56807691675761462635287905925, 11.22471308232588983253273825256, 12.06524861431498138398520264722

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