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Properties

Label 2-560-16.13-c1-0-13
Degree $2$
Conductor $560$
Sign $0.945 + 0.326i$
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  + (−0.560 − 1.29i)2-s + (0.693 + 0.693i)3-s + (−1.37 + 1.45i)4-s + (−0.707 + 0.707i)5-s + (0.511 − 1.28i)6-s − i·7-s + (2.65 + 0.963i)8-s − 2.03i·9-s + (1.31 + 0.521i)10-s + (−0.309 + 0.309i)11-s + (−1.95 + 0.0590i)12-s + (3.90 + 3.90i)13-s + (−1.29 + 0.560i)14-s − 0.980·15-s + (−0.240 − 3.99i)16-s + 0.135·17-s + ⋯
L(s)  = 1  + (−0.396 − 0.918i)2-s + (0.400 + 0.400i)3-s + (−0.685 + 0.728i)4-s + (−0.316 + 0.316i)5-s + (0.208 − 0.526i)6-s − 0.377i·7-s + (0.940 + 0.340i)8-s − 0.679i·9-s + (0.415 + 0.164i)10-s + (−0.0933 + 0.0933i)11-s + (−0.565 + 0.0170i)12-s + (1.08 + 1.08i)13-s + (−0.346 + 0.149i)14-s − 0.253·15-s + (−0.0601 − 0.998i)16-s + 0.0329·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.945 + 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25180 - 0.210033i\)
\(L(\frac12)\) \(\approx\) \(1.25180 - 0.210033i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.560 + 1.29i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + iT \)
good3 \( 1 + (-0.693 - 0.693i)T + 3iT^{2} \)
11 \( 1 + (0.309 - 0.309i)T - 11iT^{2} \)
13 \( 1 + (-3.90 - 3.90i)T + 13iT^{2} \)
17 \( 1 - 0.135T + 17T^{2} \)
19 \( 1 + (-2.90 - 2.90i)T + 19iT^{2} \)
23 \( 1 - 0.323iT - 23T^{2} \)
29 \( 1 + (-5.89 - 5.89i)T + 29iT^{2} \)
31 \( 1 - 6.72T + 31T^{2} \)
37 \( 1 + (-3.99 + 3.99i)T - 37iT^{2} \)
41 \( 1 - 3.21iT - 41T^{2} \)
43 \( 1 + (-1.58 + 1.58i)T - 43iT^{2} \)
47 \( 1 + 0.680T + 47T^{2} \)
53 \( 1 + (-1.42 + 1.42i)T - 53iT^{2} \)
59 \( 1 + (3.10 - 3.10i)T - 59iT^{2} \)
61 \( 1 + (-3.16 - 3.16i)T + 61iT^{2} \)
67 \( 1 + (9.61 + 9.61i)T + 67iT^{2} \)
71 \( 1 + 0.0650iT - 71T^{2} \)
73 \( 1 - 3.41iT - 73T^{2} \)
79 \( 1 + 9.73T + 79T^{2} \)
83 \( 1 + (11.8 + 11.8i)T + 83iT^{2} \)
89 \( 1 + 1.93iT - 89T^{2} \)
97 \( 1 - 8.46T + 97T^{2} \)
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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

−10.60779042292925239277535255405, −9.932010710158453117356818400376, −9.052165337723805757512831282194, −8.405171369805582810987329130141, −7.34414056197815627472559369089, −6.26842313832827565659251036470, −4.56106743214150665368761992093, −3.78448122335337578742942478391, −2.93649923386154918793337883653, −1.25163396394153992694255806476, 1.04211633294630316673000734203, 2.84028243256273560291022424361, 4.44373872141357332699236051651, 5.42491485244138143219237975992, 6.32891969563416248092250091878, 7.45430722352342437722535094751, 8.233840331463318259262620556134, 8.583249829295017249698512987958, 9.758957036852899029697795145707, 10.61791008766634485321122519072

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