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Properties

Label 2-561-11.3-c1-0-26
Degree $2$
Conductor $561$
Sign $0.428 + 0.903i$
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  + (2.15 − 1.56i)2-s + (0.309 + 0.951i)3-s + (1.58 − 4.87i)4-s + (2.54 + 1.84i)5-s + (2.15 + 1.56i)6-s + (0.625 − 1.92i)7-s + (−2.57 − 7.93i)8-s + (−0.809 + 0.587i)9-s + 8.39·10-s + (−2.35 + 2.33i)11-s + 5.12·12-s + (−2.41 + 1.75i)13-s + (−1.66 − 5.13i)14-s + (−0.972 + 2.99i)15-s + (−9.72 − 7.06i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (1.52 − 1.10i)2-s + (0.178 + 0.549i)3-s + (0.791 − 2.43i)4-s + (1.13 + 0.826i)5-s + (0.881 + 0.640i)6-s + (0.236 − 0.727i)7-s + (−0.911 − 2.80i)8-s + (−0.269 + 0.195i)9-s + 2.65·10-s + (−0.710 + 0.704i)11-s + 1.47·12-s + (−0.668 + 0.485i)13-s + (−0.446 − 1.37i)14-s + (−0.250 + 0.772i)15-s + (−2.43 − 1.76i)16-s + (−0.196 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(561\)    =    \(3 \cdot 11 \cdot 17\)
Sign: $0.428 + 0.903i$
Analytic conductor: \(4.47960\)
Root analytic conductor: \(2.11650\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{561} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 561,\ (\ :1/2),\ 0.428 + 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.25448 - 2.05753i\)
\(L(\frac12)\) \(\approx\) \(3.25448 - 2.05753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (2.35 - 2.33i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-2.15 + 1.56i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-2.54 - 1.84i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.625 + 1.92i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.41 - 1.75i)T + (4.01 - 12.3i)T^{2} \)
19 \( 1 + (-1.55 - 4.79i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.03T + 23T^{2} \)
29 \( 1 + (-2.81 + 8.64i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.74 + 4.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.519 - 1.59i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.90 + 5.87i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + (-1.69 - 5.20i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.44 - 1.05i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.18 - 9.78i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.98 - 6.52i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.69T + 67T^{2} \)
71 \( 1 + (3.34 + 2.43i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.75 + 5.40i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.12 + 5.90i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.164 - 0.119i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.85T + 89T^{2} \)
97 \( 1 + (0.913 - 0.663i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.35503790363995002838774303482, −10.20639941182559364464629173786, −9.682288174805689257668137046941, −7.72517568512828978741644060917, −6.46456351162473589827516899437, −5.72925912772856074590163649532, −4.67123545565991118346135305005, −3.92860276640517104367526120177, −2.62628570101165352644494122317, −1.96268011860701712285384148522, 2.19340126790506293075241349835, 3.18630590022757467629462860120, 4.97284585872126309261936583073, 5.25603481781429060204394759453, 6.15532534098510842425194183858, 6.97210441578937664752974275533, 8.262536694455545588081796891872, 8.538458588554061109944268845965, 9.908788804023229846070143781434, 11.40736622756409728431567814398

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