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Properties

Label 2-561-11.9-c1-0-0
Degree $2$
Conductor $561$
Sign $-0.894 + 0.447i$
Analytic cond. $4.47960$
Root an. cond. $2.11650$
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  + (−0.532 + 1.63i)2-s + (−0.809 + 0.587i)3-s + (−0.785 − 0.570i)4-s + (−1.21 − 3.72i)5-s + (−0.532 − 1.63i)6-s + (2.86 + 2.07i)7-s + (−1.43 + 1.04i)8-s + (0.309 − 0.951i)9-s + 6.75·10-s + (−2.46 + 2.21i)11-s + 0.970·12-s + (−0.761 + 2.34i)13-s + (−4.93 + 3.58i)14-s + (3.17 + 2.30i)15-s + (−1.54 − 4.75i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.376 + 1.15i)2-s + (−0.467 + 0.339i)3-s + (−0.392 − 0.285i)4-s + (−0.541 − 1.66i)5-s + (−0.217 − 0.669i)6-s + (1.08 + 0.785i)7-s + (−0.507 + 0.368i)8-s + (0.103 − 0.317i)9-s + 2.13·10-s + (−0.743 + 0.668i)11-s + 0.280·12-s + (−0.211 + 0.650i)13-s + (−1.31 + 0.957i)14-s + (0.819 + 0.595i)15-s + (−0.386 − 1.18i)16-s + (0.0749 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(561\)    =    \(3 \cdot 11 \cdot 17\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(4.47960\)
Root analytic conductor: \(2.11650\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{561} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 561,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109983 - 0.465976i\)
\(L(\frac12)\) \(\approx\) \(0.109983 - 0.465976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (2.46 - 2.21i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.532 - 1.63i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (1.21 + 3.72i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.86 - 2.07i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.761 - 2.34i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (2.19 - 1.59i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.32T + 23T^{2} \)
29 \( 1 + (-2.69 - 1.95i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.40 - 7.41i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (9.20 + 6.68i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (10.0 - 7.31i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + (-7.29 + 5.30i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.24 + 3.82i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.04 - 1.48i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.855 - 2.63i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 5.19T + 67T^{2} \)
71 \( 1 + (-2.41 - 7.43i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.98 + 2.17i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.126 - 0.389i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.899 - 2.76i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + (4.72 - 14.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.46713069986120207344973486253, −10.19073325042766461873008588375, −9.012187925532446319421027481896, −8.519281992447183645012128187114, −7.921962206743805192195872100235, −6.86938243336866603143134694885, −5.41734778028934195293442877374, −5.20578604605962816663599525081, −4.18640798517282962815996408696, −1.87430616483966969407687535086, 0.31185605605937120002814190528, 2.08267237410337086674215876443, 3.08331426115531813817247683782, 4.15175419404826015849559823674, 5.71124951508440403647563037881, 6.78431059019358968775745635667, 7.61756969250252504493489856418, 8.352443979247104832975144582740, 10.05484974958292513184625940455, 10.54062216710633575372185939035

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