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Properties

Label 2-1280-20.7-c1-0-19
Degree $2$
Conductor $1280$
Sign $-0.340 - 0.940i$
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  + (2.18 + 2.18i)3-s + (2.18 + 0.456i)5-s + (−1.79 + 1.79i)7-s + 6.58i·9-s + 0.913i·11-s + (1.73 − 1.73i)13-s + (3.79 + 5.79i)15-s + (3 + 3i)17-s − 3.46·19-s − 7.84·21-s + (−3.79 − 3.79i)23-s + (4.58 + 1.99i)25-s + (−7.84 + 7.84i)27-s − 5.29i·29-s − 7.58i·31-s + ⋯
L(s)  = 1  + (1.26 + 1.26i)3-s + (0.978 + 0.204i)5-s + (−0.677 + 0.677i)7-s + 2.19i·9-s + 0.275i·11-s + (0.480 − 0.480i)13-s + (0.978 + 1.49i)15-s + (0.727 + 0.727i)17-s − 0.794·19-s − 1.71·21-s + (−0.790 − 0.790i)23-s + (0.916 + 0.399i)25-s + (−1.50 + 1.50i)27-s − 0.982i·29-s − 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.340 - 0.940i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.340 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.798456375\)
\(L(\frac12)\) \(\approx\) \(2.798456375\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 0.456i)T \)
good3 \( 1 + (-2.18 - 2.18i)T + 3iT^{2} \)
7 \( 1 + (1.79 - 1.79i)T - 7iT^{2} \)
11 \( 1 - 0.913iT - 11T^{2} \)
13 \( 1 + (-1.73 + 1.73i)T - 13iT^{2} \)
17 \( 1 + (-3 - 3i)T + 17iT^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + (3.79 + 3.79i)T + 23iT^{2} \)
29 \( 1 + 5.29iT - 29T^{2} \)
31 \( 1 + 7.58iT - 31T^{2} \)
37 \( 1 + (-5.19 - 5.19i)T + 37iT^{2} \)
41 \( 1 + 1.58T + 41T^{2} \)
43 \( 1 + (0.361 + 0.361i)T + 43iT^{2} \)
47 \( 1 + (3.79 - 3.79i)T - 47iT^{2} \)
53 \( 1 + (-2.64 + 2.64i)T - 53iT^{2} \)
59 \( 1 - 5.29T + 59T^{2} \)
61 \( 1 + 6.20T + 61T^{2} \)
67 \( 1 + (-0.361 + 0.361i)T - 67iT^{2} \)
71 \( 1 - 4.41iT - 71T^{2} \)
73 \( 1 + (-8.58 + 8.58i)T - 73iT^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + (-3.10 - 3.10i)T + 83iT^{2} \)
89 \( 1 + 15.1iT - 89T^{2} \)
97 \( 1 + (8.58 + 8.58i)T + 97iT^{2} \)
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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

−9.856448261194888503710062972655, −9.299745861185072064603111908585, −8.433254259832011530356671681664, −7.908047491852167191251347181025, −6.34135469409504657404182151176, −5.79297067414972358870655135363, −4.62662554122808858753814277222, −3.72632299089845978675905314143, −2.79179106005629997867376090643, −2.09440825238449767662747074273, 1.04426950058689916098574537510, 1.98588428139939954492154962672, 3.03483134350125113680047390474, 3.84021470919061395016306762876, 5.40802673561009833059074704358, 6.49379770244957405055843620869, 6.88041619873315343301080616770, 7.80975842191694806694419250983, 8.628099872639163101763114235118, 9.281504774037207761258424687010

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