Properties

Label 2-560-16.5-c1-0-8
Degree $2$
Conductor $560$
Sign $-0.935 + 0.353i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + (−0.551 + 1.30i)2-s + (−1.16 + 1.16i)3-s + (−1.39 − 1.43i)4-s + (0.707 + 0.707i)5-s + (−0.873 − 2.15i)6-s + i·7-s + (2.63 − 1.02i)8-s + 0.295i·9-s + (−1.31 + 0.531i)10-s + (3.97 + 3.97i)11-s + (3.28 + 0.0508i)12-s + (−1.48 + 1.48i)13-s + (−1.30 − 0.551i)14-s − 1.64·15-s + (−0.123 + 3.99i)16-s − 6.33·17-s + ⋯
L(s)  = 1  + (−0.389 + 0.920i)2-s + (−0.671 + 0.671i)3-s + (−0.696 − 0.717i)4-s + (0.316 + 0.316i)5-s + (−0.356 − 0.879i)6-s + 0.377i·7-s + (0.932 − 0.361i)8-s + 0.0986i·9-s + (−0.414 + 0.167i)10-s + (1.19 + 1.19i)11-s + (0.949 + 0.0146i)12-s + (−0.413 + 0.413i)13-s + (−0.348 − 0.147i)14-s − 0.424·15-s + (−0.0309 + 0.999i)16-s − 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.935 + 0.353i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.935 + 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130670 - 0.714466i\)
\(L(\frac12)\) \(\approx\) \(0.130670 - 0.714466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.551 - 1.30i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 + (1.16 - 1.16i)T - 3iT^{2} \)
11 \( 1 + (-3.97 - 3.97i)T + 11iT^{2} \)
13 \( 1 + (1.48 - 1.48i)T - 13iT^{2} \)
17 \( 1 + 6.33T + 17T^{2} \)
19 \( 1 + (-4.80 + 4.80i)T - 19iT^{2} \)
23 \( 1 - 4.65iT - 23T^{2} \)
29 \( 1 + (4.24 - 4.24i)T - 29iT^{2} \)
31 \( 1 + 0.0294T + 31T^{2} \)
37 \( 1 + (5.73 + 5.73i)T + 37iT^{2} \)
41 \( 1 - 0.919iT - 41T^{2} \)
43 \( 1 + (3.37 + 3.37i)T + 43iT^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 + (-0.0243 - 0.0243i)T + 53iT^{2} \)
59 \( 1 + (5.91 + 5.91i)T + 59iT^{2} \)
61 \( 1 + (-1.34 + 1.34i)T - 61iT^{2} \)
67 \( 1 + (9.68 - 9.68i)T - 67iT^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 + 4.67T + 79T^{2} \)
83 \( 1 + (-4.38 + 4.38i)T - 83iT^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 - 0.578T + 97T^{2} \)
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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

−11.11073112969845674479370776130, −10.17680276508384595135447746993, −9.305754601697503431810019497795, −8.999138936915862356801377082179, −7.26682006295779150047383160063, −6.93091886510795508534369642788, −5.72124351918688857947844621781, −4.93793373884701930039831101334, −4.10101002045737889682284251348, −1.93618632378032081173149910630, 0.53549327087466945448573970470, 1.66213167720764175033259504099, 3.29379379527411350855313948600, 4.37737337560526486238369298828, 5.72791573523272358609166154248, 6.60924500293510536199883731913, 7.68770267280053047866746894273, 8.756947065397113874130509746127, 9.388534014414842518146457786488, 10.45802633761800481483510167111

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