Properties

Label 2-1280-40.19-c2-0-17
Degree $2$
Conductor $1280$
Sign $-0.998 - 0.0484i$
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  + 1.73i·3-s + (3.35 + 3.70i)5-s − 7.32·7-s + 5.97·9-s − 13.9·11-s + 12.4·13-s + (−6.43 + 5.83i)15-s − 4.07i·17-s + 14.9·19-s − 12.7i·21-s − 24.8·23-s + (−2.42 + 24.8i)25-s + 26.0i·27-s + 42.6i·29-s + 27.9i·31-s + ⋯
L(s)  = 1  + 0.579i·3-s + (0.671 + 0.740i)5-s − 1.04·7-s + 0.664·9-s − 1.27·11-s + 0.954·13-s + (−0.429 + 0.389i)15-s − 0.239i·17-s + 0.788·19-s − 0.606i·21-s − 1.08·23-s + (−0.0968 + 0.995i)25-s + 0.964i·27-s + 1.47i·29-s + 0.902i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0484i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.998 - 0.0484i$
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: \((2,\ 1280,\ (\ :1),\ -0.998 - 0.0484i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.020337362\)
\(L(\frac12)\) \(\approx\) \(1.020337362\)
\(L(2)\) 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 \( 1 + (-3.35 - 3.70i)T \)
good3 \( 1 - 1.73iT - 9T^{2} \)
7 \( 1 + 7.32T + 49T^{2} \)
11 \( 1 + 13.9T + 121T^{2} \)
13 \( 1 - 12.4T + 169T^{2} \)
17 \( 1 + 4.07iT - 289T^{2} \)
19 \( 1 - 14.9T + 361T^{2} \)
23 \( 1 + 24.8T + 529T^{2} \)
29 \( 1 - 42.6iT - 841T^{2} \)
31 \( 1 - 27.9iT - 961T^{2} \)
37 \( 1 - 12.4T + 1.36e3T^{2} \)
41 \( 1 + 75.4T + 1.68e3T^{2} \)
43 \( 1 + 20.5iT - 1.84e3T^{2} \)
47 \( 1 - 31.8T + 2.20e3T^{2} \)
53 \( 1 + 88.3T + 2.80e3T^{2} \)
59 \( 1 + 64.3T + 3.48e3T^{2} \)
61 \( 1 + 44.1iT - 3.72e3T^{2} \)
67 \( 1 + 85.0iT - 4.48e3T^{2} \)
71 \( 1 - 44.9iT - 5.04e3T^{2} \)
73 \( 1 - 51.8iT - 5.32e3T^{2} \)
79 \( 1 - 145. iT - 6.24e3T^{2} \)
83 \( 1 + 100. iT - 6.88e3T^{2} \)
89 \( 1 + 15.4T + 7.92e3T^{2} \)
97 \( 1 - 0.492iT - 9.40e3T^{2} \)
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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

−9.887786052100030135799088098000, −9.406092171208743813295521975949, −8.322110178768296139701890915582, −7.24948787574812051969488182540, −6.60335791987604843159159026448, −5.69573295495128509293828195857, −4.90247344438050059791013681388, −3.51237280010537274430080734251, −3.05464407208035217456574311845, −1.64428205737838344083442837530, 0.28311394196370456879471261462, 1.53155058554462220036643509878, 2.59238036999936683609633808834, 3.82215938857932296765205158533, 4.89249023491087296019676129406, 6.03258473856470279694162586151, 6.29493080607685336061748585423, 7.61276817977264508578816565487, 8.107015441956339035437715184422, 9.180336642341250413357587081726

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