Properties

Label 2-1280-8.5-c3-0-49
Degree $2$
Conductor $1280$
Sign $0.707 + 0.707i$
Analytic cond. $75.5224$
Root an. cond. $8.69036$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.32i·3-s − 5i·5-s + 18.9·7-s − 13.0·9-s − 12.6i·11-s + 38i·13-s − 31.6·15-s + 34·17-s + 101. i·19-s − 120. i·21-s + 82.2·23-s − 25·25-s − 88.5i·27-s + 270i·29-s + 341.·31-s + ⋯
L(s)  = 1  − 1.21i·3-s − 0.447i·5-s + 1.02·7-s − 0.481·9-s − 0.346i·11-s + 0.810i·13-s − 0.544·15-s + 0.485·17-s + 1.22i·19-s − 1.24i·21-s + 0.745·23-s − 0.200·25-s − 0.631i·27-s + 1.72i·29-s + 1.97·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(75.5224\)
Root analytic conductor: \(8.69036\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :3/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.716825981\)
\(L(\frac12)\) \(\approx\) \(2.716825981\)
\(L(\frac{5}{2})\) 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 \( 1 + 5iT \)
good3 \( 1 + 6.32iT - 27T^{2} \)
7 \( 1 - 18.9T + 343T^{2} \)
11 \( 1 + 12.6iT - 1.33e3T^{2} \)
13 \( 1 - 38iT - 2.19e3T^{2} \)
17 \( 1 - 34T + 4.91e3T^{2} \)
19 \( 1 - 101. iT - 6.85e3T^{2} \)
23 \( 1 - 82.2T + 1.21e4T^{2} \)
29 \( 1 - 270iT - 2.43e4T^{2} \)
31 \( 1 - 341.T + 2.97e4T^{2} \)
37 \( 1 + 206iT - 5.06e4T^{2} \)
41 \( 1 - 270T + 6.89e4T^{2} \)
43 \( 1 - 537. iT - 7.95e4T^{2} \)
47 \( 1 - 132.T + 1.03e5T^{2} \)
53 \( 1 - 258iT - 1.48e5T^{2} \)
59 \( 1 + 75.8iT - 2.05e5T^{2} \)
61 \( 1 + 250iT - 2.26e5T^{2} \)
67 \( 1 - 815. iT - 3.00e5T^{2} \)
71 \( 1 + 645.T + 3.57e5T^{2} \)
73 \( 1 - 1.07e3T + 3.89e5T^{2} \)
79 \( 1 - 278.T + 4.93e5T^{2} \)
83 \( 1 - 1.10e3iT - 5.71e5T^{2} \)
89 \( 1 + 890T + 7.04e5T^{2} \)
97 \( 1 + 254T + 9.12e5T^{2} \)
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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

−8.961191641023146651683363534619, −8.171411304203537848776108986114, −7.69559969783873962568737410740, −6.79551862338365513620816475566, −5.98123965967583634683919885294, −5.01686517577959828682513994859, −4.10801582112110541595441152344, −2.69418925867076843712065486338, −1.48762615716734369709373136345, −1.06890589330679983074728970496, 0.78024702920185381145338010395, 2.36328466462289636218519248887, 3.32378646258497278403151611240, 4.46125160074986236842289185107, 4.88824420254756755105151917381, 5.86877200099169365062526465349, 7.01698997222897868735342726722, 7.88263974997316005730614854069, 8.634666035991470286309201339659, 9.611564893064050047887554089583

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