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Properties

Label 2-560-140.79-c0-0-0
Degree $2$
Conductor $560$
Sign $0.444 - 0.895i$
Analytic cond. $0.279476$
Root an. cond. $0.528655$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 1.5i)3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)7-s + (−1 + 1.73i)9-s + 1.73·15-s + (−1.5 − 0.866i)21-s + (0.866 − 1.5i)23-s + (−0.499 − 0.866i)25-s − 1.73·27-s − 29-s + 0.999i·35-s + 41-s − 1.73·43-s + (1 + 1.73i)45-s + (0.499 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.866 + 1.5i)3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)7-s + (−1 + 1.73i)9-s + 1.73·15-s + (−1.5 − 0.866i)21-s + (0.866 − 1.5i)23-s + (−0.499 − 0.866i)25-s − 1.73·27-s − 29-s + 0.999i·35-s + 41-s − 1.73·43-s + (1 + 1.73i)45-s + (0.499 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(0.279476\)
Root analytic conductor: \(0.528655\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :0),\ 0.444 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.138815759\)
\(L(\frac12)\) \(\approx\) \(1.138815759\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + 1.73T + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - 1.73T + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.73246574393042764256793151734, −10.07718578632670281953175912325, −9.153348060492659599121548564511, −9.007692664274587538701087269020, −7.983030576075986604938129003869, −6.40252308270542119281461890621, −5.30827537287805132407581586852, −4.52521211089171466808079360822, −3.44145577226018232264669468676, −2.39048707730289601442247771640, 1.60945424140335401371850451392, 2.84049898887639670279557810562, 3.57956012057656180467308860837, 5.68149038148112441402137672218, 6.62316452598821405034664398445, 7.18252502849567000973489341711, 7.86614928068396116444057771731, 9.114138848507732387425526765548, 9.726653748447647576375262447434, 10.86812253055021900634849981756

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