Properties

Degree $2$
Conductor $128$
Sign $i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82i·3-s − 5.00·9-s − 2.82i·11-s + 6·17-s + 8.48i·19-s + 5·25-s + 5.65i·27-s − 8.00·33-s − 6·41-s + 8.48i·43-s − 7·49-s − 16.9i·51-s + 24·57-s − 14.1i·59-s + 8.48i·67-s + ⋯
L(s)  = 1  − 1.63i·3-s − 1.66·9-s − 0.852i·11-s + 1.45·17-s + 1.94i·19-s + 25-s + 1.08i·27-s − 1.39·33-s − 0.937·41-s + 1.29i·43-s − 49-s − 2.37i·51-s + 3.17·57-s − 1.84i·59-s + 1.03i·67-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $i$
Motivic weight: \(1\)
Character: $\chi_{128} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.748881 - 0.748881i\)
\(L(\frac12)\) \(\approx\) \(0.748881 - 0.748881i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.82iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 8.48iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 8.48iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−12.92089563107896110519093354136, −12.29818754311778415065996876260, −11.36515988012176547866224364384, −10.00565361958677993380557336950, −8.380648284949726943323366842492, −7.76869240934951829307603994212, −6.51950332514357795625674257850, −5.58550133389113302414045527469, −3.23595485619640913873366119897, −1.38179389167272920808910458982, 3.08634036933586957896715177110, 4.48789758656622716181904592878, 5.35336554189979674456025177029, 7.08971988569328979813016041294, 8.664242019433138992834480113785, 9.588916288933705161532427497691, 10.35406548918393499584234425732, 11.29975411628232970704714828288, 12.44943601301639991206556802577, 13.82686932364188423095463472958

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