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Properties

Label 2-2e7-8.5-c3-0-3
Degree $2$
Conductor $128$
Sign $0.707 - 0.707i$
Analytic cond. $7.55224$
Root an. cond. $2.74813$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·5-s + 27·9-s + 92i·13-s + 94·17-s + 109·25-s + 284i·29-s − 396i·37-s − 230·41-s + 108i·45-s − 343·49-s − 572i·53-s − 468i·61-s − 368·65-s − 1.09e3·73-s + 729·81-s + ⋯
L(s)  = 1  + 0.357i·5-s + 9-s + 1.96i·13-s + 1.34·17-s + 0.871·25-s + 1.81i·29-s − 1.75i·37-s − 0.876·41-s + 0.357i·45-s − 49-s − 1.48i·53-s − 0.982i·61-s − 0.702·65-s − 1.76·73-s + 0.999·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(7.55224\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.57612 + 0.652850i\)
\(L(\frac12)\) \(\approx\) \(1.57612 + 0.652850i\)
\(L(\frac{5}{2})\) 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 \( 1 - 27T^{2} \)
5 \( 1 - 4iT - 125T^{2} \)
7 \( 1 + 343T^{2} \)
11 \( 1 - 1.33e3T^{2} \)
13 \( 1 - 92iT - 2.19e3T^{2} \)
17 \( 1 - 94T + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 284iT - 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 + 396iT - 5.06e4T^{2} \)
41 \( 1 + 230T + 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 572iT - 1.48e5T^{2} \)
59 \( 1 - 2.05e5T^{2} \)
61 \( 1 + 468iT - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 - 1.67e3T + 7.04e5T^{2} \)
97 \( 1 + 594T + 9.12e5T^{2} \)
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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

−12.90736782961264201904883385231, −12.02125191394609853393009838453, −10.91463585366786182503598448256, −9.867588944926950366979321077115, −8.896593904414148534767087127391, −7.36912480561218050578042127831, −6.60777374400933069046152240068, −4.94421065723907476703441997808, −3.60300395994520040456512855161, −1.65573843803848776582738571451, 1.02517492739886578989439019968, 3.15780485715801365597521995324, 4.74578903472573108525590239224, 5.94473080992444743154752892682, 7.48428172509871663596900133783, 8.309045790252605057850777275456, 9.852509159261334332860912531266, 10.40303101307729163173774827611, 11.90764113632520561511652251883, 12.78484732429561978649617370521

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