Properties

Label 2-560-35.12-c1-0-14
Degree $2$
Conductor $560$
Sign $0.528 + 0.848i$
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  + (0.133 − 0.5i)3-s + (0.133 − 2.23i)5-s + (2.5 + 0.866i)7-s + (2.36 + 1.36i)9-s + (−1.36 − 2.36i)11-s + (−2 − 2i)13-s + (−1.09 − 0.366i)15-s + (3.73 + i)17-s + (−0.366 + 0.633i)19-s + (0.767 − 1.13i)21-s + (−0.0358 − 0.133i)23-s + (−4.96 − 0.598i)25-s + (2.09 − 2.09i)27-s − 3i·29-s + (6.46 − 3.73i)31-s + ⋯
L(s)  = 1  + (0.0773 − 0.288i)3-s + (0.0599 − 0.998i)5-s + (0.944 + 0.327i)7-s + (0.788 + 0.455i)9-s + (−0.411 − 0.713i)11-s + (−0.554 − 0.554i)13-s + (−0.283 − 0.0945i)15-s + (0.905 + 0.242i)17-s + (−0.0839 + 0.145i)19-s + (0.167 − 0.247i)21-s + (−0.00748 − 0.0279i)23-s + (−0.992 − 0.119i)25-s + (0.403 − 0.403i)27-s − 0.557i·29-s + (1.16 − 0.670i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.528 + 0.848i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.528 + 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46950 - 0.816006i\)
\(L(\frac12)\) \(\approx\) \(1.46950 - 0.816006i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.133 + 2.23i)T \)
7 \( 1 + (-2.5 - 0.866i)T \)
good3 \( 1 + (-0.133 + 0.5i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.36 + 2.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 + 2i)T + 13iT^{2} \)
17 \( 1 + (-3.73 - i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.366 - 0.633i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0358 + 0.133i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (-6.46 + 3.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.73 - 1.26i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.46iT - 41T^{2} \)
43 \( 1 + (2.83 - 2.83i)T - 43iT^{2} \)
47 \( 1 + (2.36 + 8.83i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.83 - 1.83i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.09 - 7.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.33 - 0.767i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.86 - 10.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.26T + 71T^{2} \)
73 \( 1 + (3.46 - 12.9i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.83 + 1.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.09 - 2.09i)T + 83iT^{2} \)
89 \( 1 + (0.330 - 0.571i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.92 - 5.92i)T - 97iT^{2} \)
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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

−10.48037414935866491008926133464, −9.857774485482531784108617952347, −8.524426299720178714808624405741, −8.140855207872584245379539498002, −7.27375612590516772429018853883, −5.73541475843463463203343388238, −5.12271736934294828556066785657, −4.08737456179851799988518545782, −2.38552181202392545504218212042, −1.10255016746565741358781149159, 1.70258881407282085513826207091, 3.09642107649895656094075037443, 4.31324408447124397099924329877, 5.14043367461765199068711254715, 6.62309151778698760282845053007, 7.26738058745607335418650563103, 8.065530739955170105230045598795, 9.403688850494770143448678140730, 10.12889241264250841260349528626, 10.70459029094506903220613024829

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