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Properties

Label 2-1280-80.69-c1-0-9
Degree $2$
Conductor $1280$
Sign $0.990 + 0.134i$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.53 − 1.53i)3-s + (−0.569 − 2.16i)5-s − 4.20·7-s + 1.73i·9-s + (4.38 + 4.38i)11-s + (2.17 + 2.17i)13-s + (−2.44 + 4.20i)15-s + 2.75i·17-s + (−1.93 + 1.93i)19-s + (6.46 + 6.46i)21-s − 3.37·23-s + (−4.35 + 2.46i)25-s + (−1.95 + 1.95i)27-s + (4.46 − 4.46i)29-s + 1.03·31-s + ⋯
L(s)  = 1  + (−0.888 − 0.888i)3-s + (−0.254 − 0.966i)5-s − 1.58·7-s + 0.577i·9-s + (1.32 + 1.32i)11-s + (0.603 + 0.603i)13-s + (−0.632 + 1.08i)15-s + 0.668i·17-s + (−0.443 + 0.443i)19-s + (1.41 + 1.41i)21-s − 0.704·23-s + (−0.870 + 0.492i)25-s + (−0.375 + 0.375i)27-s + (0.828 − 0.828i)29-s + 0.185·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.990 + 0.134i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.990 + 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7960372449\)
\(L(\frac12)\) \(\approx\) \(0.7960372449\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.569 + 2.16i)T \)
good3 \( 1 + (1.53 + 1.53i)T + 3iT^{2} \)
7 \( 1 + 4.20T + 7T^{2} \)
11 \( 1 + (-4.38 - 4.38i)T + 11iT^{2} \)
13 \( 1 + (-2.17 - 2.17i)T + 13iT^{2} \)
17 \( 1 - 2.75iT - 17T^{2} \)
19 \( 1 + (1.93 - 1.93i)T - 19iT^{2} \)
23 \( 1 + 3.37T + 23T^{2} \)
29 \( 1 + (-4.46 + 4.46i)T - 29iT^{2} \)
31 \( 1 - 1.03T + 31T^{2} \)
37 \( 1 + (-7.53 + 7.53i)T - 37iT^{2} \)
41 \( 1 - 3.26iT - 41T^{2} \)
43 \( 1 + (-4.91 + 4.91i)T - 43iT^{2} \)
47 \( 1 - 0.301iT - 47T^{2} \)
53 \( 1 + (4.93 - 4.93i)T - 53iT^{2} \)
59 \( 1 + (-4.76 - 4.76i)T + 59iT^{2} \)
61 \( 1 + (2 - 2i)T - 61iT^{2} \)
67 \( 1 + (4.31 + 4.31i)T + 67iT^{2} \)
71 \( 1 - 3.58iT - 71T^{2} \)
73 \( 1 - 4.77T + 73T^{2} \)
79 \( 1 + 3.86T + 79T^{2} \)
83 \( 1 + (-3.48 - 3.48i)T + 83iT^{2} \)
89 \( 1 + 7.46iT - 89T^{2} \)
97 \( 1 - 15.8iT - 97T^{2} \)
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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

−9.480990276941809550904137129356, −9.065460179725407081602058766092, −7.87397111758551560142378413978, −6.96025246558953652595203398905, −6.25919513117778916657373420284, −5.91064686013362610416758192261, −4.34548037527158071661660844897, −3.84581645563392772352836362441, −1.99346024736612021392321323270, −0.902453075224477609667881819638, 0.51927937875501291540270822466, 2.94935948542847078771790029637, 3.51177074392895729039496778739, 4.40730078163340312702245239151, 5.79437812889038442989091477441, 6.29181739937522701249243492936, 6.79806122961330903051544427202, 8.138193576337505276011091278227, 9.144911388593923246971597059555, 9.866195627340502866609121158841

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