Properties

Label 2-1280-40.19-c2-0-27
Degree $2$
Conductor $1280$
Sign $0.891 + 0.453i$
Analytic cond. $34.8774$
Root an. cond. $5.90571$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.30i·3-s + (−4.75 + 1.54i)5-s − 0.206·7-s − 19.1·9-s + 15.0·11-s − 11.6·13-s + (8.20 + 25.2i)15-s + 18.1i·17-s + 19.3·19-s + 1.09i·21-s − 27.2·23-s + (20.2 − 14.7i)25-s + 53.6i·27-s + 44.4i·29-s + 20.3i·31-s + ⋯
L(s)  = 1  − 1.76i·3-s + (−0.950 + 0.309i)5-s − 0.0295·7-s − 2.12·9-s + 1.36·11-s − 0.899·13-s + (0.547 + 1.68i)15-s + 1.07i·17-s + 1.02·19-s + 0.0521i·21-s − 1.18·23-s + (0.808 − 0.588i)25-s + 1.98i·27-s + 1.53i·29-s + 0.657i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(34.8774\)
Root analytic conductor: \(5.90571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1),\ 0.891 + 0.453i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.293049065\)
\(L(\frac12)\) \(\approx\) \(1.293049065\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (4.75 - 1.54i)T \)
good3 \( 1 + 5.30iT - 9T^{2} \)
7 \( 1 + 0.206T + 49T^{2} \)
11 \( 1 - 15.0T + 121T^{2} \)
13 \( 1 + 11.6T + 169T^{2} \)
17 \( 1 - 18.1iT - 289T^{2} \)
19 \( 1 - 19.3T + 361T^{2} \)
23 \( 1 + 27.2T + 529T^{2} \)
29 \( 1 - 44.4iT - 841T^{2} \)
31 \( 1 - 20.3iT - 961T^{2} \)
37 \( 1 - 18.1T + 1.36e3T^{2} \)
41 \( 1 - 32.3T + 1.68e3T^{2} \)
43 \( 1 + 4.06iT - 1.84e3T^{2} \)
47 \( 1 + 5.37T + 2.20e3T^{2} \)
53 \( 1 - 79.1T + 2.80e3T^{2} \)
59 \( 1 - 83.3T + 3.48e3T^{2} \)
61 \( 1 + 36.7iT - 3.72e3T^{2} \)
67 \( 1 - 4.51iT - 4.48e3T^{2} \)
71 \( 1 + 41.6iT - 5.04e3T^{2} \)
73 \( 1 - 41.5iT - 5.32e3T^{2} \)
79 \( 1 + 15.5iT - 6.24e3T^{2} \)
83 \( 1 - 50.9iT - 6.88e3T^{2} \)
89 \( 1 + 10.8T + 7.92e3T^{2} \)
97 \( 1 - 12.1iT - 9.40e3T^{2} \)
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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

−9.146303565975132321345333389040, −8.318212295347034830704712002869, −7.68452485249327293690566965967, −6.95687506580066788822758382209, −6.49591210049396404011230068166, −5.44798004020163335786672005078, −4.05207070189527987484598795486, −3.09341457717730061835016582946, −1.87920078305425698084940935354, −0.883245132619155590146366686346, 0.51096498228739439687272468220, 2.67396810497745037467489117824, 3.79093380079508861177427471045, 4.23843647651802452118093172167, 5.02421414298336131793580554004, 5.95423261774884475924642591481, 7.22457999631391838691168503513, 8.074501095991379046592948013341, 9.007903211990938031185534723045, 9.645027265416245190444531766975

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