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Properties

Label 2-1280-40.19-c2-0-39
Degree $2$
Conductor $1280$
Sign $-0.248 - 0.968i$
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  + 2.75i·3-s + (−2.54 + 4.30i)5-s + 3.84·7-s + 1.41·9-s − 6.19·11-s + 16.1·13-s + (−11.8 − 7.01i)15-s + 5.20i·17-s + 36.2·19-s + 10.6i·21-s + 22.0·23-s + (−12.0 − 21.9i)25-s + 28.6i·27-s + 20.0i·29-s − 26.4i·31-s + ⋯
L(s)  = 1  + 0.918i·3-s + (−0.509 + 0.860i)5-s + 0.549·7-s + 0.157·9-s − 0.562·11-s + 1.23·13-s + (−0.789 − 0.467i)15-s + 0.306i·17-s + 1.90·19-s + 0.504i·21-s + 0.958·23-s + (−0.480 − 0.876i)25-s + 1.06i·27-s + 0.690i·29-s − 0.852i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.248 - 0.968i$
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: \((2,\ 1280,\ (\ :1),\ -0.248 - 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.215807366\)
\(L(\frac12)\) \(\approx\) \(2.215807366\)
\(L(2)\) 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 \( 1 + (2.54 - 4.30i)T \)
good3 \( 1 - 2.75iT - 9T^{2} \)
7 \( 1 - 3.84T + 49T^{2} \)
11 \( 1 + 6.19T + 121T^{2} \)
13 \( 1 - 16.1T + 169T^{2} \)
17 \( 1 - 5.20iT - 289T^{2} \)
19 \( 1 - 36.2T + 361T^{2} \)
23 \( 1 - 22.0T + 529T^{2} \)
29 \( 1 - 20.0iT - 841T^{2} \)
31 \( 1 + 26.4iT - 961T^{2} \)
37 \( 1 - 69.3T + 1.36e3T^{2} \)
41 \( 1 + 11.6T + 1.68e3T^{2} \)
43 \( 1 - 25.8iT - 1.84e3T^{2} \)
47 \( 1 - 66.1T + 2.20e3T^{2} \)
53 \( 1 + 39.5T + 2.80e3T^{2} \)
59 \( 1 + 27.7T + 3.48e3T^{2} \)
61 \( 1 - 54.1iT - 3.72e3T^{2} \)
67 \( 1 + 107. iT - 4.48e3T^{2} \)
71 \( 1 + 70.7iT - 5.04e3T^{2} \)
73 \( 1 - 37.4iT - 5.32e3T^{2} \)
79 \( 1 + 97.6iT - 6.24e3T^{2} \)
83 \( 1 - 126. iT - 6.88e3T^{2} \)
89 \( 1 + 133.T + 7.92e3T^{2} \)
97 \( 1 + 6.40iT - 9.40e3T^{2} \)
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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

−9.753518226334784334384846747483, −9.070589201735199114548412644193, −7.936426480355601139966877896892, −7.47461158393538656442861124912, −6.37355019250288312269483947144, −5.39085213583214218128734282565, −4.49847821680769772523495663655, −3.61763373648666297354780994524, −2.87271072677316796165710519301, −1.19771029690457182847580116795, 0.828865936131464766320126838151, 1.43300233914330201948610735681, 2.91687088358048661067811036367, 4.11497535415414476400612336919, 5.05996815417382311225610977367, 5.84087052375494077173270886994, 7.01356821253573184126155058072, 7.68384387186509278804519988343, 8.231258817222356372041672150355, 9.079445150398687587152651407490

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