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Properties

Label 2-560-35.3-c1-0-6
Degree $2$
Conductor $560$
Sign $-0.454 - 0.890i$
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  + (0.617 + 2.30i)3-s + (2.21 − 0.313i)5-s + (−2.53 − 0.755i)7-s + (−2.33 + 1.34i)9-s + (−2.18 + 3.78i)11-s + (−4.36 + 4.36i)13-s + (2.09 + 4.91i)15-s + (1.63 − 0.438i)17-s + (3.56 + 6.17i)19-s + (0.175 − 6.31i)21-s + (1.31 − 4.91i)23-s + (4.80 − 1.38i)25-s + (0.510 + 0.510i)27-s + 1.33i·29-s + (1.90 + 1.09i)31-s + ⋯
L(s)  = 1  + (0.356 + 1.33i)3-s + (0.990 − 0.140i)5-s + (−0.958 − 0.285i)7-s + (−0.778 + 0.449i)9-s + (−0.659 + 1.14i)11-s + (−1.21 + 1.21i)13-s + (0.539 + 1.26i)15-s + (0.396 − 0.106i)17-s + (0.818 + 1.41i)19-s + (0.0382 − 1.37i)21-s + (0.274 − 1.02i)23-s + (0.960 − 0.277i)25-s + (0.0982 + 0.0982i)27-s + 0.247i·29-s + (0.341 + 0.197i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.454 - 0.890i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.803314 + 1.31138i\)
\(L(\frac12)\) \(\approx\) \(0.803314 + 1.31138i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.313i)T \)
7 \( 1 + (2.53 + 0.755i)T \)
good3 \( 1 + (-0.617 - 2.30i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.36 - 4.36i)T - 13iT^{2} \)
17 \( 1 + (-1.63 + 0.438i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.56 - 6.17i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.31 + 4.91i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.33iT - 29T^{2} \)
31 \( 1 + (-1.90 - 1.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.839 - 0.224i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.69iT - 41T^{2} \)
43 \( 1 + (3.40 + 3.40i)T + 43iT^{2} \)
47 \( 1 + (-2.63 + 9.84i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.02 + 0.541i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.56 + 4.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.21 + 3.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.90 + 7.11i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.31T + 71T^{2} \)
73 \( 1 + (3.82 + 14.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.82 + 2.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.272 - 0.272i)T - 83iT^{2} \)
89 \( 1 + (1.79 + 3.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.325 + 0.325i)T + 97iT^{2} \)
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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

−10.44716464641933285383147973737, −9.965699666388269684998200202485, −9.677965690801744029646362983539, −8.816687847919839133395864739608, −7.40836857307934692018179299543, −6.48976375734052962778887507710, −5.19747934459870514152903433943, −4.54557764199307927964542479370, −3.34335206385920462056531566878, −2.16279861246589467374952231410, 0.837918092586494702768807273489, 2.62993544237216863426590098330, 2.95760146495854943799773874755, 5.32960540964598957922693332217, 5.89677415217053656269523475031, 6.98724817240613303372356490774, 7.60670333927614685294548699833, 8.660048277528547996306146147348, 9.598669683039724296775070253880, 10.31335743613730860636418173292

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