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Properties

Label 2-561-11.9-c1-0-14
Degree $2$
Conductor $561$
Sign $0.831 - 0.555i$
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  + (−0.0304 + 0.0937i)2-s + (0.809 − 0.587i)3-s + (1.61 + 1.16i)4-s + (0.171 + 0.527i)5-s + (0.0304 + 0.0937i)6-s + (2.92 + 2.12i)7-s + (−0.318 + 0.231i)8-s + (0.309 − 0.951i)9-s − 0.0547·10-s + (−2.17 + 2.50i)11-s + 1.99·12-s + (2.09 − 6.46i)13-s + (−0.288 + 0.209i)14-s + (0.448 + 0.326i)15-s + (1.21 + 3.74i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.0215 + 0.0663i)2-s + (0.467 − 0.339i)3-s + (0.805 + 0.584i)4-s + (0.0766 + 0.235i)5-s + (0.0124 + 0.0382i)6-s + (1.10 + 0.803i)7-s + (−0.112 + 0.0817i)8-s + (0.103 − 0.317i)9-s − 0.0173·10-s + (−0.654 + 0.755i)11-s + 0.574·12-s + (0.582 − 1.79i)13-s + (−0.0771 + 0.0560i)14-s + (0.115 + 0.0842i)15-s + (0.304 + 0.937i)16-s + (0.0749 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(561\)    =    \(3 \cdot 11 \cdot 17\)
Sign: $0.831 - 0.555i$
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: \((2,\ 561,\ (\ :1/2),\ 0.831 - 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06704 + 0.626519i\)
\(L(\frac12)\) \(\approx\) \(2.06704 + 0.626519i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 - 2.50i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.0304 - 0.0937i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.171 - 0.527i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.92 - 2.12i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-2.09 + 6.46i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (5.31 - 3.85i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 8.75T + 23T^{2} \)
29 \( 1 + (-5.66 - 4.11i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.69 + 5.23i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.60 + 4.79i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.14 + 3.74i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.50T + 43T^{2} \)
47 \( 1 + (1.20 - 0.877i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.97 + 9.15i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.18 - 3.03i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.92 - 5.92i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 1.66T + 67T^{2} \)
71 \( 1 + (4.05 + 12.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.0104 - 0.00761i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.05 - 9.40i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.74 + 11.5i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.22T + 89T^{2} \)
97 \( 1 + (-0.152 + 0.469i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.62046569883124517518789344476, −10.35837665324901960470683783159, −8.577093099301625920600582657962, −8.154804045684754656697530173981, −7.56963537795955357000537362161, −6.30099882692591044702360281538, −5.49559569500468768893176591270, −3.99629172799246092939046154632, −2.66871160104453734432026725651, −1.93826814390569712853843736238, 1.39224807247684555084498721438, 2.51838831976091782895007091488, 4.11371473298408124152473326752, 4.89853076097778644722174956991, 6.20318593392692784205453711863, 7.04856851999088171619855675639, 8.151570781138022468127677192260, 8.823026298248657589096092126670, 9.980457386735472245856845517148, 10.78107011683175529659539258800

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