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Properties

Label 2-560-16.5-c1-0-38
Degree $2$
Conductor $560$
Sign $0.827 + 0.561i$
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  + (1.39 − 0.204i)2-s + (0.989 − 0.989i)3-s + (1.91 − 0.573i)4-s + (0.707 + 0.707i)5-s + (1.18 − 1.58i)6-s + i·7-s + (2.56 − 1.19i)8-s + 1.04i·9-s + (1.13 + 0.844i)10-s + (−3.71 − 3.71i)11-s + (1.32 − 2.46i)12-s + (2.81 − 2.81i)13-s + (0.204 + 1.39i)14-s + 1.39·15-s + (3.34 − 2.19i)16-s + 0.895·17-s + ⋯
L(s)  = 1  + (0.989 − 0.144i)2-s + (0.571 − 0.571i)3-s + (0.958 − 0.286i)4-s + (0.316 + 0.316i)5-s + (0.482 − 0.648i)6-s + 0.377i·7-s + (0.906 − 0.422i)8-s + 0.346i·9-s + (0.358 + 0.267i)10-s + (−1.12 − 1.12i)11-s + (0.383 − 0.711i)12-s + (0.780 − 0.780i)13-s + (0.0547 + 0.373i)14-s + 0.361·15-s + (0.835 − 0.549i)16-s + 0.217·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.827 + 0.561i$
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: \((2,\ 560,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.11013 - 0.956107i\)
\(L(\frac12)\) \(\approx\) \(3.11013 - 0.956107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.204i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 + (-0.989 + 0.989i)T - 3iT^{2} \)
11 \( 1 + (3.71 + 3.71i)T + 11iT^{2} \)
13 \( 1 + (-2.81 + 2.81i)T - 13iT^{2} \)
17 \( 1 - 0.895T + 17T^{2} \)
19 \( 1 + (3.33 - 3.33i)T - 19iT^{2} \)
23 \( 1 - 6.38iT - 23T^{2} \)
29 \( 1 + (-1.18 + 1.18i)T - 29iT^{2} \)
31 \( 1 + 6.25T + 31T^{2} \)
37 \( 1 + (4.00 + 4.00i)T + 37iT^{2} \)
41 \( 1 - 2.72iT - 41T^{2} \)
43 \( 1 + (8.61 + 8.61i)T + 43iT^{2} \)
47 \( 1 - 3.21T + 47T^{2} \)
53 \( 1 + (3.27 + 3.27i)T + 53iT^{2} \)
59 \( 1 + (-7.51 - 7.51i)T + 59iT^{2} \)
61 \( 1 + (-0.859 + 0.859i)T - 61iT^{2} \)
67 \( 1 + (7.29 - 7.29i)T - 67iT^{2} \)
71 \( 1 + 5.87iT - 71T^{2} \)
73 \( 1 - 14.4iT - 73T^{2} \)
79 \( 1 + 4.29T + 79T^{2} \)
83 \( 1 + (10.3 - 10.3i)T - 83iT^{2} \)
89 \( 1 + 0.769iT - 89T^{2} \)
97 \( 1 - 16.7T + 97T^{2} \)
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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

−10.75915156178284727182524155081, −10.19411870237550990405538833090, −8.614191535316748283225565600514, −7.936923888491368359698263695154, −7.03474414848591365615691057394, −5.70679924910914206944150530283, −5.46736372211225828136312218117, −3.64814058464693368275080416520, −2.83926043046309627279202321589, −1.75034897471215952847492450214, 2.00767867186496748041033271634, 3.20268074458150502742500458797, 4.35713063676457368010935408528, 4.89312038390887560277272954979, 6.27164075170201630564840179673, 7.03221937627735345900818443965, 8.191267383269008834524887033052, 9.056395350974723078457416122802, 10.15763755068836776353780214431, 10.74233130078170046999400736967

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