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Properties

Label 2-1280-8.5-c1-0-14
Degree $2$
Conductor $1280$
Sign $0.707 - 0.707i$
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  + 2i·3-s − i·5-s + 2·7-s − 9-s − 4i·11-s + 6i·13-s + 2·15-s + 2·17-s − 8i·19-s + 4i·21-s + 6·23-s − 25-s + 4i·27-s + 2i·29-s + 4·31-s + ⋯
L(s)  = 1  + 1.15i·3-s − 0.447i·5-s + 0.755·7-s − 0.333·9-s − 1.20i·11-s + 1.66i·13-s + 0.516·15-s + 0.485·17-s − 1.83i·19-s + 0.872i·21-s + 1.25·23-s − 0.200·25-s + 0.769i·27-s + 0.371i·29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.927469047\)
\(L(\frac12)\) \(\approx\) \(1.927469047\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 - 2iT - 3T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 - 6iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−9.548542620219817254205937044250, −9.046866256524912021517874461995, −8.513715621080274122741456904179, −7.33199107908077772433713637004, −6.41557191448445407871384308461, −5.17736275527103570162766340915, −4.71579075765337062072571809757, −3.89817626275689786006102587650, −2.74380996083893799449863223262, −1.14278176642425696447911837790, 1.08571203973696756141475815507, 2.07445675364317249813953668184, 3.17743772980011364424921909151, 4.48420504069430678262865452901, 5.54024007330661594377694783736, 6.31521360589688984426753952956, 7.42789027444500383537495659354, 7.68613461182798119360937015743, 8.378850205366672997921174603243, 9.776200963334991338265089865452

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