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Properties

Label 2-1280-16.13-c1-0-16
Degree $2$
Conductor $1280$
Sign $-0.130 - 0.991i$
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  + (2.07 + 2.07i)3-s + (0.707 − 0.707i)5-s + 4.34i·7-s + 5.59i·9-s + (2.66 − 2.66i)11-s + (−0.482 − 0.482i)13-s + 2.93·15-s + 0.353·17-s + (5.39 + 5.39i)19-s + (−9.00 + 9.00i)21-s − 6.62i·23-s − 1.00i·25-s + (−5.38 + 5.38i)27-s + (−3.42 − 3.42i)29-s − 0.635·31-s + ⋯
L(s)  = 1  + (1.19 + 1.19i)3-s + (0.316 − 0.316i)5-s + 1.64i·7-s + 1.86i·9-s + (0.803 − 0.803i)11-s + (−0.133 − 0.133i)13-s + 0.757·15-s + 0.0856·17-s + (1.23 + 1.23i)19-s + (−1.96 + 1.96i)21-s − 1.38i·23-s − 0.200i·25-s + (−1.03 + 1.03i)27-s + (−0.636 − 0.636i)29-s − 0.114·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.130 - 0.991i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.130 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.737314119\)
\(L(\frac12)\) \(\approx\) \(2.737314119\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-2.07 - 2.07i)T + 3iT^{2} \)
7 \( 1 - 4.34iT - 7T^{2} \)
11 \( 1 + (-2.66 + 2.66i)T - 11iT^{2} \)
13 \( 1 + (0.482 + 0.482i)T + 13iT^{2} \)
17 \( 1 - 0.353T + 17T^{2} \)
19 \( 1 + (-5.39 - 5.39i)T + 19iT^{2} \)
23 \( 1 + 6.62iT - 23T^{2} \)
29 \( 1 + (3.42 + 3.42i)T + 29iT^{2} \)
31 \( 1 + 0.635T + 31T^{2} \)
37 \( 1 + (3.13 - 3.13i)T - 37iT^{2} \)
41 \( 1 + 2.33iT - 41T^{2} \)
43 \( 1 + (7.59 - 7.59i)T - 43iT^{2} \)
47 \( 1 + 5.41T + 47T^{2} \)
53 \( 1 + (4.49 - 4.49i)T - 53iT^{2} \)
59 \( 1 + (-9.46 + 9.46i)T - 59iT^{2} \)
61 \( 1 + (-6.21 - 6.21i)T + 61iT^{2} \)
67 \( 1 + (-0.362 - 0.362i)T + 67iT^{2} \)
71 \( 1 + 7.32iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 - 8.13T + 79T^{2} \)
83 \( 1 + (8.41 + 8.41i)T + 83iT^{2} \)
89 \( 1 - 4.55iT - 89T^{2} \)
97 \( 1 - 2.17T + 97T^{2} \)
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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

−9.623423789791033478173779959851, −9.147720746876663428045816298544, −8.439638392531215138598110372712, −8.006895694718641404572179255914, −6.35838832419830317930312745852, −5.54311331089296146887413073034, −4.76501479499459263514360994231, −3.60203241833893060439928007872, −2.93566423092497240917682499811, −1.86965675106843992932303434630, 1.08834351388124470029722074253, 1.92529604606402138159913831497, 3.23600918664047299206790054253, 3.87691603190168490866805384974, 5.21532420812280893479773892588, 6.79153497449269558944504945619, 7.07328872487170244310064120651, 7.50927483167651060336182068716, 8.546906384525315582150332380122, 9.522272547808785641091309728707

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