Properties

Degree $2$
Conductor $128$
Sign $0.999 + 0.00911i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.10 + 2.67i)3-s + (2.95 − 7.13i)5-s + (4.18 − 4.18i)7-s + (0.437 − 0.437i)9-s + (−1.42 + 3.44i)11-s + (8.39 + 20.2i)13-s + 22.3·15-s − 1.73i·17-s + (−14.2 + 5.90i)19-s + (15.8 + 6.55i)21-s + (−15.1 − 15.1i)23-s + (−24.5 − 24.5i)25-s + (25.7 + 10.6i)27-s + (−6.74 + 2.79i)29-s − 31.1i·31-s + ⋯
L(s)  = 1  + (0.369 + 0.891i)3-s + (0.591 − 1.42i)5-s + (0.597 − 0.597i)7-s + (0.0486 − 0.0486i)9-s + (−0.129 + 0.313i)11-s + (0.646 + 1.55i)13-s + 1.49·15-s − 0.101i·17-s + (−0.749 + 0.310i)19-s + (0.753 + 0.312i)21-s + (−0.658 − 0.658i)23-s + (−0.980 − 0.980i)25-s + (0.952 + 0.394i)27-s + (−0.232 + 0.0962i)29-s − 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00911i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.999 + 0.00911i$
Motivic weight: \(2\)
Character: $\chi_{128} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ 0.999 + 0.00911i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.79071 - 0.00816358i\)
\(L(\frac12)\) \(\approx\) \(1.79071 - 0.00816358i\)
\(L(2)\) 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 \( 1 + (-1.10 - 2.67i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (-2.95 + 7.13i)T + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (-4.18 + 4.18i)T - 49iT^{2} \)
11 \( 1 + (1.42 - 3.44i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-8.39 - 20.2i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + 1.73iT - 289T^{2} \)
19 \( 1 + (14.2 - 5.90i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (15.1 + 15.1i)T + 529iT^{2} \)
29 \( 1 + (6.74 - 2.79i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 31.1iT - 961T^{2} \)
37 \( 1 + (-5.30 + 12.7i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (18.5 - 18.5i)T - 1.68e3iT^{2} \)
43 \( 1 + (31.0 - 75.0i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 16.2T + 2.20e3T^{2} \)
53 \( 1 + (29.0 + 12.0i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (34.1 + 14.1i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (68.7 - 28.4i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (10.5 + 25.3i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-32.2 + 32.2i)T - 5.04e3iT^{2} \)
73 \( 1 + (28.5 - 28.5i)T - 5.32e3iT^{2} \)
79 \( 1 + 22.4T + 6.24e3T^{2} \)
83 \( 1 + (-123. + 51.0i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-61.0 - 61.0i)T + 7.92e3iT^{2} \)
97 \( 1 + 69.9T + 9.40e3T^{2} \)
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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

−13.20707780256729278363321460388, −12.15532765516333744592597522709, −10.89691828428841053341250052822, −9.709410235798188573644922229675, −9.081732131611935679936167730053, −8.081151080945638521327478757738, −6.32756758786208874663579396582, −4.68804095083223463270416927718, −4.17480341826786219908315986132, −1.61491796097820291919692198053, 1.96403188754745299606385152216, 3.15981837477322915428545465912, 5.52497170984922195733718335433, 6.58216764925669947320709807249, 7.68755244947260677408669033191, 8.583084752050917865856845277866, 10.27982797416872262813182970330, 10.87994432336108466409393753093, 12.19688659659919658027086371427, 13.33424959864729458050132988419

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