Properties

Label 2-2e7-8.3-c4-0-3
Degree $2$
Conductor $128$
Sign $-0.707 - 0.707i$
Analytic cond. $13.2313$
Root an. cond. $3.63749$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.54·3-s + 16.8i·5-s + 40.7i·7-s − 60.3·9-s − 104.·11-s − 13.4i·13-s + 76.4i·15-s + 73.3·17-s − 502.·19-s + 185. i·21-s + 973. i·23-s + 341.·25-s − 642.·27-s − 1.25e3i·29-s + 1.08e3i·31-s + ⋯
L(s)  = 1  + 0.504·3-s + 0.673i·5-s + 0.832i·7-s − 0.745·9-s − 0.860·11-s − 0.0796i·13-s + 0.339i·15-s + 0.253·17-s − 1.39·19-s + 0.420i·21-s + 1.84i·23-s + 0.546·25-s − 0.880·27-s − 1.49i·29-s + 1.13i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(13.2313\)
Root analytic conductor: \(3.63749\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.439402 + 1.06081i\)
\(L(\frac12)\) \(\approx\) \(0.439402 + 1.06081i\)
\(L(3)\) 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 \( 1 - 4.54T + 81T^{2} \)
5 \( 1 - 16.8iT - 625T^{2} \)
7 \( 1 - 40.7iT - 2.40e3T^{2} \)
11 \( 1 + 104.T + 1.46e4T^{2} \)
13 \( 1 + 13.4iT - 2.85e4T^{2} \)
17 \( 1 - 73.3T + 8.35e4T^{2} \)
19 \( 1 + 502.T + 1.30e5T^{2} \)
23 \( 1 - 973. iT - 2.79e5T^{2} \)
29 \( 1 + 1.25e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.08e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.95e3iT - 1.87e6T^{2} \)
41 \( 1 + 194.T + 2.82e6T^{2} \)
43 \( 1 + 1.80e3T + 3.41e6T^{2} \)
47 \( 1 - 627. iT - 4.87e6T^{2} \)
53 \( 1 - 3.01e3iT - 7.89e6T^{2} \)
59 \( 1 - 5.69e3T + 1.21e7T^{2} \)
61 \( 1 + 6.07e3iT - 1.38e7T^{2} \)
67 \( 1 - 3.51e3T + 2.01e7T^{2} \)
71 \( 1 + 3.33e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.81e3T + 2.83e7T^{2} \)
79 \( 1 + 5.46e3iT - 3.89e7T^{2} \)
83 \( 1 - 3.94e3T + 4.74e7T^{2} \)
89 \( 1 + 700.T + 6.27e7T^{2} \)
97 \( 1 + 1.27e4T + 8.85e7T^{2} \)
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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

−13.14851407443888731269733729007, −11.93270080506665883839254746299, −10.98817928193442639471628091850, −9.872833265401268182766107599930, −8.677667820696202310789715171139, −7.84597806317573069962252307721, −6.38400420441561701459762295699, −5.23750817578164595535056123251, −3.31400986350117699113432182742, −2.28751573720800879920663195028, 0.42060002777338522111677524400, 2.43141178235876721482946076966, 4.03935520347276450139177822501, 5.32258024227476524723140165064, 6.84708867356574076179444513921, 8.209358437889809540827269922734, 8.799648227075516307777964734767, 10.23985321125536797815909604862, 11.06020909571071416308873441417, 12.55287243192009047610335256557

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