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Properties

Label 2-561-11.9-c1-0-18
Degree $2$
Conductor $561$
Sign $-0.0732 + 0.997i$
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  + (0.569 − 1.75i)2-s + (−0.809 + 0.587i)3-s + (−1.13 − 0.821i)4-s + (−0.208 − 0.642i)5-s + (0.569 + 1.75i)6-s + (1.38 + 1.00i)7-s + (0.897 − 0.651i)8-s + (0.309 − 0.951i)9-s − 1.24·10-s + (0.643 + 3.25i)11-s + 1.39·12-s + (1.25 − 3.84i)13-s + (2.54 − 1.85i)14-s + (0.546 + 0.396i)15-s + (−1.49 − 4.60i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (0.402 − 1.23i)2-s + (−0.467 + 0.339i)3-s + (−0.565 − 0.410i)4-s + (−0.0933 − 0.287i)5-s + (0.232 + 0.715i)6-s + (0.522 + 0.379i)7-s + (0.317 − 0.230i)8-s + (0.103 − 0.317i)9-s − 0.393·10-s + (0.193 + 0.981i)11-s + 0.403·12-s + (0.346 − 1.06i)13-s + (0.680 − 0.494i)14-s + (0.141 + 0.102i)15-s + (−0.374 − 1.15i)16-s + (0.0749 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0732 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0732 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(561\)    =    \(3 \cdot 11 \cdot 17\)
Sign: $-0.0732 + 0.997i$
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: \((2,\ 561,\ (\ :1/2),\ -0.0732 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19146 - 1.28214i\)
\(L(\frac12)\) \(\approx\) \(1.19146 - 1.28214i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.643 - 3.25i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.569 + 1.75i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.208 + 0.642i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.38 - 1.00i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.25 + 3.84i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (-5.32 + 3.86i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.57T + 23T^{2} \)
29 \( 1 + (0.748 + 0.543i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.588 - 1.81i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.07 + 2.96i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.41 + 1.02i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.85T + 43T^{2} \)
47 \( 1 + (3.49 - 2.53i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.61 - 8.04i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.34 - 4.61i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.984 + 3.03i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 0.966T + 67T^{2} \)
71 \( 1 + (2.68 + 8.25i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.57 - 6.23i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.87 - 11.9i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.63 - 14.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 6.13T + 89T^{2} \)
97 \( 1 + (5.26 - 16.1i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.73621521797759039442910305363, −9.960706953304754936103895094098, −9.135000857723478510399923504157, −7.927527151262511179222600964924, −6.89619448342497940127584445035, −5.41577454513011575068811114870, −4.77123736308884298541733105574, −3.72084955356268084892261741317, −2.54295395528740011302008181970, −1.12908542661254266941755240109, 1.53047335884931994693293981832, 3.56752169914479200426217846347, 4.76431159437814759887230764251, 5.66226591061682546206579142395, 6.45641830195665150931648341430, 7.23953031644734541278182944805, 7.964773579884060600968341614984, 8.891576954535445495082032980960, 10.20552895962022134219431778050, 11.30017613545702790468818982818

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