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Properties

Label 2-560-35.9-c1-0-4
Degree $2$
Conductor $560$
Sign $0.305 - 0.952i$
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  + (0.277 − 0.159i)3-s + (−2.00 − 0.986i)5-s + (1.50 + 2.17i)7-s + (−1.44 + 2.50i)9-s + (2.08 + 3.60i)11-s − 2.89i·13-s + (−0.713 + 0.0477i)15-s + (−3.72 + 2.15i)17-s + (−0.979 + 1.69i)19-s + (0.765 + 0.361i)21-s + (2.23 + 1.28i)23-s + (3.05 + 3.95i)25-s + 1.88i·27-s + 5.96·29-s + (4.71 + 8.16i)31-s + ⋯
L(s)  = 1  + (0.159 − 0.0923i)3-s + (−0.897 − 0.441i)5-s + (0.569 + 0.821i)7-s + (−0.482 + 0.836i)9-s + (0.628 + 1.08i)11-s − 0.803i·13-s + (−0.184 + 0.0123i)15-s + (−0.903 + 0.521i)17-s + (−0.224 + 0.389i)19-s + (0.167 + 0.0788i)21-s + (0.465 + 0.268i)23-s + (0.610 + 0.791i)25-s + 0.363i·27-s + 1.10·29-s + (0.846 + 1.46i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.305 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.961168 + 0.700664i\)
\(L(\frac12)\) \(\approx\) \(0.961168 + 0.700664i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.00 + 0.986i)T \)
7 \( 1 + (-1.50 - 2.17i)T \)
good3 \( 1 + (-0.277 + 0.159i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.08 - 3.60i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.89iT - 13T^{2} \)
17 \( 1 + (3.72 - 2.15i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.979 - 1.69i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.23 - 1.28i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.96T + 29T^{2} \)
31 \( 1 + (-4.71 - 8.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.48 - 3.16i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.19T + 41T^{2} \)
43 \( 1 + 4.73iT - 43T^{2} \)
47 \( 1 + (3.84 + 2.21i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.37 - 3.67i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.61 - 2.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.94 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.1 - 5.84i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.58T + 71T^{2} \)
73 \( 1 + (-8.57 + 4.95i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.41 + 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.75iT - 83T^{2} \)
89 \( 1 + (1.50 - 2.61i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.414iT - 97T^{2} \)
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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

−11.04275563664686309009363013811, −10.14245561199253771985176030788, −8.814464784901460692668134543833, −8.398761947106119642068530524749, −7.58201425623957135618764653701, −6.45671186311366019962838970645, −5.09472666027638932713861759665, −4.54480155514044943531705146096, −3.05702291719201204459723702481, −1.70799931608828354247294946827, 0.70140897630312426983587144118, 2.81057620299281450398531319110, 3.91385880514316412698664487048, 4.60946191189549970637948932426, 6.34700963716670963168942708626, 6.84709341300814464734666421546, 8.048360205759138721977717634149, 8.701990837246236076419852520510, 9.633683503916117355341655641003, 10.89658456364342666597328465125

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