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Properties

Label 2-1280-20.19-c2-0-7
Degree $2$
Conductor $1280$
Sign $-0.870 + 0.492i$
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  − 4.79·3-s + (2.46 + 4.35i)5-s − 7.67·7-s + 13.9·9-s + 0.472i·11-s + 4.10i·13-s + (−11.7 − 20.8i)15-s + 2.26i·17-s + 26.3i·19-s + 36.7·21-s − 9.73·23-s + (−12.8 + 21.4i)25-s − 23.6·27-s + 41.6·29-s + 22.0i·31-s + ⋯
L(s)  = 1  − 1.59·3-s + (0.492 + 0.870i)5-s − 1.09·7-s + 1.54·9-s + 0.0429i·11-s + 0.316i·13-s + (−0.785 − 1.38i)15-s + 0.133i·17-s + 1.38i·19-s + 1.75·21-s − 0.423·23-s + (−0.515 + 0.856i)25-s − 0.877·27-s + 1.43·29-s + 0.710i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.870 + 0.492i$
Analytic conductor: \(34.8774\)
Root analytic conductor: \(5.90571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1),\ -0.870 + 0.492i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3067133240\)
\(L(\frac12)\) \(\approx\) \(0.3067133240\)
\(L(2)\) 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 \( 1 + (-2.46 - 4.35i)T \)
good3 \( 1 + 4.79T + 9T^{2} \)
7 \( 1 + 7.67T + 49T^{2} \)
11 \( 1 - 0.472iT - 121T^{2} \)
13 \( 1 - 4.10iT - 169T^{2} \)
17 \( 1 - 2.26iT - 289T^{2} \)
19 \( 1 - 26.3iT - 361T^{2} \)
23 \( 1 + 9.73T + 529T^{2} \)
29 \( 1 - 41.6T + 841T^{2} \)
31 \( 1 - 22.0iT - 961T^{2} \)
37 \( 1 - 51.7iT - 1.36e3T^{2} \)
41 \( 1 + 15.0T + 1.68e3T^{2} \)
43 \( 1 - 9.31T + 1.84e3T^{2} \)
47 \( 1 - 8.76T + 2.20e3T^{2} \)
53 \( 1 + 39.9iT - 2.80e3T^{2} \)
59 \( 1 - 77.1iT - 3.48e3T^{2} \)
61 \( 1 - 14.7T + 3.72e3T^{2} \)
67 \( 1 - 75.8T + 4.48e3T^{2} \)
71 \( 1 + 81.0iT - 5.04e3T^{2} \)
73 \( 1 - 83.4iT - 5.32e3T^{2} \)
79 \( 1 + 100. iT - 6.24e3T^{2} \)
83 \( 1 - 0.266T + 6.88e3T^{2} \)
89 \( 1 + 85.5T + 7.92e3T^{2} \)
97 \( 1 + 99.1iT - 9.40e3T^{2} \)
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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

−10.22544941056983831601807046714, −9.570830918335228529945505801164, −8.252683431969507224634032356901, −7.01727768692093989864962399265, −6.48747699073874541627972230722, −6.00609448678599181540200194107, −5.15289662270378436849822943698, −3.97426116245003054306469680431, −2.90446218591683610878102467658, −1.42080505782837339854375053703, 0.14589065260250129368925923395, 0.926707556219228540981947144482, 2.55550416673745760558561021275, 4.07244898619606806693604512304, 4.98182668600786747466833362235, 5.63717800640193894684185545317, 6.38796095476630668890765896156, 6.97271721677211727502767673068, 8.240795511066946454339140153557, 9.305010230571312834626736239780

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