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Properties

Label 2-560-16.5-c1-0-15
Degree $2$
Conductor $560$
Sign $-0.913 - 0.407i$
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  + (1.17 + 0.793i)2-s + (−1.25 + 1.25i)3-s + (0.740 + 1.85i)4-s + (0.707 + 0.707i)5-s + (−2.47 + 0.474i)6-s + i·7-s + (−0.607 + 2.76i)8-s − 0.163i·9-s + (0.266 + 1.38i)10-s + (−0.651 − 0.651i)11-s + (−3.26 − 1.40i)12-s + (1.11 − 1.11i)13-s + (−0.793 + 1.17i)14-s − 1.77·15-s + (−2.90 + 2.75i)16-s − 0.603·17-s + ⋯
L(s)  = 1  + (0.827 + 0.561i)2-s + (−0.726 + 0.726i)3-s + (0.370 + 0.928i)4-s + (0.316 + 0.316i)5-s + (−1.00 + 0.193i)6-s + 0.377i·7-s + (−0.214 + 0.976i)8-s − 0.0545i·9-s + (0.0843 + 0.439i)10-s + (−0.196 − 0.196i)11-s + (−0.943 − 0.405i)12-s + (0.308 − 0.308i)13-s + (−0.212 + 0.312i)14-s − 0.459·15-s + (−0.725 + 0.688i)16-s − 0.146·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.913 - 0.407i$
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: \((2,\ 560,\ (\ :1/2),\ -0.913 - 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.364838 + 1.71461i\)
\(L(\frac12)\) \(\approx\) \(0.364838 + 1.71461i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 - 0.793i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 + (1.25 - 1.25i)T - 3iT^{2} \)
11 \( 1 + (0.651 + 0.651i)T + 11iT^{2} \)
13 \( 1 + (-1.11 + 1.11i)T - 13iT^{2} \)
17 \( 1 + 0.603T + 17T^{2} \)
19 \( 1 + (0.0481 - 0.0481i)T - 19iT^{2} \)
23 \( 1 - 3.32iT - 23T^{2} \)
29 \( 1 + (-4.73 + 4.73i)T - 29iT^{2} \)
31 \( 1 + 6.15T + 31T^{2} \)
37 \( 1 + (-3.26 - 3.26i)T + 37iT^{2} \)
41 \( 1 + 5.83iT - 41T^{2} \)
43 \( 1 + (-6.87 - 6.87i)T + 43iT^{2} \)
47 \( 1 + 5.47T + 47T^{2} \)
53 \( 1 + (-8.02 - 8.02i)T + 53iT^{2} \)
59 \( 1 + (-3.27 - 3.27i)T + 59iT^{2} \)
61 \( 1 + (-8.00 + 8.00i)T - 61iT^{2} \)
67 \( 1 + (0.200 - 0.200i)T - 67iT^{2} \)
71 \( 1 - 1.91iT - 71T^{2} \)
73 \( 1 - 7.86iT - 73T^{2} \)
79 \( 1 - 1.51T + 79T^{2} \)
83 \( 1 + (-11.4 + 11.4i)T - 83iT^{2} \)
89 \( 1 - 1.79iT - 89T^{2} \)
97 \( 1 - 4.28T + 97T^{2} \)
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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

−11.22285445379784202693535343235, −10.49865545767106832243460096832, −9.464950256856865701863108058553, −8.334123209625309879894002795475, −7.40072647949217644065544694156, −6.20118971867747177362194038721, −5.63483953111658640565896141267, −4.78407408350643943762340703666, −3.72384672320807618815802747294, −2.44912451070714869474575387657, 0.854741643446121077075409549480, 2.11451358827851362207199885437, 3.65255202004716115555077725758, 4.79494393756821778810316285582, 5.71685821309096505973971892914, 6.55638866167040528205513752136, 7.25768405846436119651912682485, 8.750516637880766872531723846995, 9.773502288783869832730826380511, 10.69790229191465488909052379229

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