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Properties

Label 2-660-11.4-c1-0-7
Degree $2$
Conductor $660$
Sign $-0.868 + 0.495i$
Analytic cond. $5.27012$
Root an. cond. $2.29567$
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  + (0.309 − 0.951i)3-s + (0.809 − 0.587i)5-s + (−1.26 − 3.88i)7-s + (−0.809 − 0.587i)9-s + (−2.91 + 1.57i)11-s + (−5.72 − 4.15i)13-s + (−0.309 − 0.951i)15-s + (−0.657 + 0.477i)17-s + (−2.56 + 7.90i)19-s − 4.08·21-s + 7.02·23-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (−2.19 − 6.75i)29-s + (3.69 + 2.68i)31-s + ⋯
L(s)  = 1  + (0.178 − 0.549i)3-s + (0.361 − 0.262i)5-s + (−0.477 − 1.46i)7-s + (−0.269 − 0.195i)9-s + (−0.880 + 0.474i)11-s + (−1.58 − 1.15i)13-s + (−0.0797 − 0.245i)15-s + (−0.159 + 0.115i)17-s + (−0.588 + 1.81i)19-s − 0.891·21-s + 1.46·23-s + (0.0618 − 0.190i)25-s + (−0.155 + 0.113i)27-s + (−0.407 − 1.25i)29-s + (0.663 + 0.482i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.868 + 0.495i$
Analytic conductor: \(5.27012\)
Root analytic conductor: \(2.29567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{660} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 660,\ (\ :1/2),\ -0.868 + 0.495i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.250867 - 0.947126i\)
\(L(\frac12)\) \(\approx\) \(0.250867 - 0.947126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (2.91 - 1.57i)T \)
good7 \( 1 + (1.26 + 3.88i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (5.72 + 4.15i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.657 - 0.477i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.56 - 7.90i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 7.02T + 23T^{2} \)
29 \( 1 + (2.19 + 6.75i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.69 - 2.68i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.971 + 2.98i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.96 + 6.05i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.35T + 43T^{2} \)
47 \( 1 + (-0.858 + 2.64i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.62 + 4.08i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.34 + 4.12i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.12 + 5.90i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 3.26T + 67T^{2} \)
71 \( 1 + (-7.22 + 5.24i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.649 - 1.99i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.1 + 7.38i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-11.3 + 8.21i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.800T + 89T^{2} \)
97 \( 1 + (3.90 + 2.83i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.22673736509979233811020707939, −9.529892073097433812390974695850, −8.104482274567585114200653987785, −7.58188913030954815015254334676, −6.81001205244832711105034695849, −5.63933315593772925493218954916, −4.64428920457192212488393812472, −3.38185414976393642645305937306, −2.14525285064813378127067258252, −0.47612401528920793054719059676, 2.51178543525038440811087364420, 2.82617156390251462375813959524, 4.70724179557354990811234237279, 5.25922246049036101597313944470, 6.43571660276663246032637117516, 7.26597170540978299018466360594, 8.676964908195636836285332403558, 9.178773989087708831410257639518, 9.807611518445197611767921415646, 10.93944783372753094187295142853

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