Properties

Label 2-560-5.4-c1-0-13
Degree $2$
Conductor $560$
Sign $-0.447 + 0.894i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + (−2 − i)5-s i·7-s − 6·9-s − 3·11-s i·13-s + (3 − 6i)15-s − 5i·17-s − 8·19-s + 3·21-s + 2i·23-s + (3 + 4i)25-s − 9i·27-s + 29-s + 2·31-s + ⋯
L(s)  = 1  + 1.73i·3-s + (−0.894 − 0.447i)5-s − 0.377i·7-s − 2·9-s − 0.904·11-s − 0.277i·13-s + (0.774 − 1.54i)15-s − 1.21i·17-s − 1.83·19-s + 0.654·21-s + 0.417i·23-s + (0.600 + 0.800i)25-s − 1.73i·27-s + 0.185·29-s + 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
7 \( 1 + iT \)
good3 \( 1 - 3iT - 3T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 11iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 10iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 7T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 8T + 89T^{2} \)
97 \( 1 - 3iT - 97T^{2} \)
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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

−10.39122221674609264460421941462, −9.813340107754439583591978097041, −8.673555457906657883500618249836, −8.199690345447704683467608904797, −6.90625967382598233767942439212, −5.36047721508849218345547444582, −4.72136855038995652583907868073, −3.91813131858336118609679447058, −2.89313021270953584683833638981, 0, 1.90029477991754274331086169687, 2.90182251378309740686117137120, 4.40118322555487914118174305000, 6.00644602808738444314514610434, 6.54283079933294675299992408879, 7.58129235973462725917552446622, 8.129436152953682363906960379319, 8.835539964443643956614724393747, 10.58804003912029222761714093689

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