Properties

Degree $2$
Conductor $128$
Sign $-0.489 - 0.872i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.564i)3-s + (6.58 + 15.8i)5-s + (−14.5 + 14.5i)7-s + (−17.5 − 17.5i)9-s + (−34.4 + 14.2i)11-s + (15.3 − 37.1i)13-s + 25.3i·15-s + 103. i·17-s + (−12.4 + 29.9i)19-s + (−28.0 + 11.6i)21-s + (72.3 + 72.3i)23-s + (−120. + 120. i)25-s + (−29.2 − 70.5i)27-s + (23.9 + 9.90i)29-s + 124.·31-s + ⋯
L(s)  = 1  + (0.262 + 0.108i)3-s + (0.588 + 1.42i)5-s + (−0.785 + 0.785i)7-s + (−0.650 − 0.650i)9-s + (−0.944 + 0.391i)11-s + (0.328 − 0.792i)13-s + 0.436i·15-s + 1.47i·17-s + (−0.149 + 0.361i)19-s + (−0.291 + 0.120i)21-s + (0.656 + 0.656i)23-s + (−0.966 + 0.966i)25-s + (−0.208 − 0.503i)27-s + (0.153 + 0.0634i)29-s + 0.722·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.489 - 0.872i$
Motivic weight: \(3\)
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.489 - 0.872i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.681586 + 1.16422i\)
\(L(\frac12)\) \(\approx\) \(0.681586 + 1.16422i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.36 - 0.564i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-6.58 - 15.8i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (14.5 - 14.5i)T - 343iT^{2} \)
11 \( 1 + (34.4 - 14.2i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-15.3 + 37.1i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 103. iT - 4.91e3T^{2} \)
19 \( 1 + (12.4 - 29.9i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-72.3 - 72.3i)T + 1.21e4iT^{2} \)
29 \( 1 + (-23.9 - 9.90i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 124.T + 2.97e4T^{2} \)
37 \( 1 + (-18.0 - 43.5i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (45.1 + 45.1i)T + 6.89e4iT^{2} \)
43 \( 1 + (-457. + 189. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 582. iT - 1.03e5T^{2} \)
53 \( 1 + (-395. + 163. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (142. + 344. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (34.8 + 14.4i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (196. + 81.4i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-520. + 520. i)T - 3.57e5iT^{2} \)
73 \( 1 + (582. + 582. i)T + 3.89e5iT^{2} \)
79 \( 1 + 157. iT - 4.93e5T^{2} \)
83 \( 1 + (54.5 - 131. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (272. - 272. i)T - 7.04e5iT^{2} \)
97 \( 1 - 788.T + 9.12e5T^{2} \)
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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

−13.20034182269929837182761679271, −12.31597441410525911758547413820, −10.87457186572583842002939682878, −10.18910886151327367574941335255, −9.128092946138558000568048164118, −7.84260058959697824564828905492, −6.37159878568443959212228648019, −5.76042949963763294721724058886, −3.38422840719244477327964259466, −2.53450037443163660957469721745, 0.65273658823773463938822981810, 2.64535885938283581896574180904, 4.56231194661571956040882462552, 5.60745376384868623172369197679, 7.11304610973792403024703738743, 8.459187048077988215157283114351, 9.216375705617876548259616017328, 10.32486314408303697329896741862, 11.54691772335094607054075946380, 12.88761428370351040394977980831

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