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Properties

Label 2-660-44.43-c1-0-28
Degree $2$
Conductor $660$
Sign $0.487 + 0.873i$
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  + (−0.934 + 1.06i)2-s + i·3-s + (−0.254 − 1.98i)4-s − 5-s + (−1.06 − 0.934i)6-s − 1.04·7-s + (2.34 + 1.58i)8-s − 9-s + (0.934 − 1.06i)10-s + (−1.23 − 3.07i)11-s + (1.98 − 0.254i)12-s + 6.64i·13-s + (0.972 − 1.10i)14-s − i·15-s + (−3.87 + 1.01i)16-s − 4.77i·17-s + ⋯
L(s)  = 1  + (−0.660 + 0.750i)2-s + 0.577i·3-s + (−0.127 − 0.991i)4-s − 0.447·5-s + (−0.433 − 0.381i)6-s − 0.393·7-s + (0.828 + 0.559i)8-s − 0.333·9-s + (0.295 − 0.335i)10-s + (−0.372 − 0.928i)11-s + (0.572 − 0.0735i)12-s + 1.84i·13-s + (0.259 − 0.295i)14-s − 0.258i·15-s + (−0.967 + 0.252i)16-s − 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.487 + 0.873i$
Analytic conductor: \(5.27012\)
Root analytic conductor: \(2.29567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{660} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 660,\ (\ :1/2),\ 0.487 + 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327684 - 0.192388i\)
\(L(\frac12)\) \(\approx\) \(0.327684 - 0.192388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.934 - 1.06i)T \)
3 \( 1 - iT \)
5 \( 1 + T \)
11 \( 1 + (1.23 + 3.07i)T \)
good7 \( 1 + 1.04T + 7T^{2} \)
13 \( 1 - 6.64iT - 13T^{2} \)
17 \( 1 + 4.77iT - 17T^{2} \)
19 \( 1 + 3.82T + 19T^{2} \)
23 \( 1 + 8.15iT - 23T^{2} \)
29 \( 1 + 1.35iT - 29T^{2} \)
31 \( 1 + 6.87iT - 31T^{2} \)
37 \( 1 - 3.24T + 37T^{2} \)
41 \( 1 + 5.90iT - 41T^{2} \)
43 \( 1 + 1.55T + 43T^{2} \)
47 \( 1 - 2.30iT - 47T^{2} \)
53 \( 1 + 1.01T + 53T^{2} \)
59 \( 1 - 0.222iT - 59T^{2} \)
61 \( 1 + 7.59iT - 61T^{2} \)
67 \( 1 + 4.98iT - 67T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + 3.42iT - 73T^{2} \)
79 \( 1 + 2.92T + 79T^{2} \)
83 \( 1 + 1.64T + 83T^{2} \)
89 \( 1 - 4.49T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−10.21403539420091539229155815921, −9.303134658233458529901752586562, −8.769479265314300748050601898058, −7.87972901185147261282013248319, −6.76117204112178208585157317784, −6.16038260517775873033603116066, −4.86870569417585116869138373882, −4.08656711590127349039473330745, −2.43420181640834440623383266478, −0.25808740371490662553411387495, 1.45574041244602893448731862112, 2.83060177758485886632387265923, 3.74148675749517718845239586619, 5.13150837644606412118351811892, 6.44744887309975072031739995781, 7.57326386706628001430927680399, 7.987459259214046187362241208886, 8.910124262770964248068133092878, 10.06278806817947243450404006095, 10.52344427437072737026485831627

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