Properties

Degree $2$
Conductor $128$
Sign $-0.377 - 0.925i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.56 − 1.89i)3-s + (1.37 + 3.30i)5-s + (6.14 − 6.14i)7-s + (−1.79 − 1.79i)9-s + (−17.2 + 7.14i)11-s + (−25.9 + 62.7i)13-s − 17.7i·15-s + 87.5i·17-s + (−48.8 + 117. i)19-s + (−39.7 + 16.4i)21-s + (55.7 + 55.7i)23-s + (79.3 − 79.3i)25-s + (55.9 + 134. i)27-s + (−114. − 47.2i)29-s − 229.·31-s + ⋯
L(s)  = 1  + (−0.879 − 0.364i)3-s + (0.122 + 0.295i)5-s + (0.331 − 0.331i)7-s + (−0.0665 − 0.0665i)9-s + (−0.472 + 0.195i)11-s + (−0.554 + 1.33i)13-s − 0.304i·15-s + 1.24i·17-s + (−0.589 + 1.42i)19-s + (−0.412 + 0.170i)21-s + (0.505 + 0.505i)23-s + (0.634 − 0.634i)25-s + (0.398 + 0.962i)27-s + (−0.730 − 0.302i)29-s − 1.33·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.377 - 0.925i$
Motivic weight: \(3\)
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.377 - 0.925i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.330846 + 0.492280i\)
\(L(\frac12)\) \(\approx\) \(0.330846 + 0.492280i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.56 + 1.89i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-1.37 - 3.30i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (-6.14 + 6.14i)T - 343iT^{2} \)
11 \( 1 + (17.2 - 7.14i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (25.9 - 62.7i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 87.5iT - 4.91e3T^{2} \)
19 \( 1 + (48.8 - 117. i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-55.7 - 55.7i)T + 1.21e4iT^{2} \)
29 \( 1 + (114. + 47.2i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 229.T + 2.97e4T^{2} \)
37 \( 1 + (123. + 298. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-111. - 111. i)T + 6.89e4iT^{2} \)
43 \( 1 + (76.5 - 31.7i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 367. iT - 1.03e5T^{2} \)
53 \( 1 + (244. - 101. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-183. - 442. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-524. - 217. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (393. + 162. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-354. + 354. i)T - 3.57e5iT^{2} \)
73 \( 1 + (22.2 + 22.2i)T + 3.89e5iT^{2} \)
79 \( 1 - 396. iT - 4.93e5T^{2} \)
83 \( 1 + (-410. + 990. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-170. + 170. i)T - 7.04e5iT^{2} \)
97 \( 1 + 1.72e3T + 9.12e5T^{2} \)
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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

−12.91929206337215379799019618950, −12.17018766478191320073743234161, −11.13331040201710113038909745720, −10.36957309954482377665589743541, −8.991451152454276245847192273843, −7.58165039129959325128867825200, −6.53541919995726997465115551929, −5.50365988114107593685169211216, −3.98937379322205847613394840151, −1.80181015784927710257478356730, 0.32971439258420958348719613579, 2.76869368660602743888250225825, 5.01281655259034038182286487696, 5.32961020923782425887239685631, 6.98187386513565495628708789177, 8.317214510406595350543854938445, 9.481166495216801740707874089291, 10.76048912238801406044457461879, 11.28934631851355965461296544262, 12.53251607993802953894290045473

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