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Properties

Label 2-1140-19.6-c1-0-11
Degree $2$
Conductor $1140$
Sign $-0.258 + 0.965i$
Analytic cond. $9.10294$
Root an. cond. $3.01710$
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  + (−0.939 − 0.342i)3-s + (0.173 + 0.984i)5-s + (2.18 − 3.78i)7-s + (0.766 + 0.642i)9-s + (−1.46 − 2.53i)11-s + (1.97 − 0.718i)13-s + (0.173 − 0.984i)15-s + (−4.23 + 3.55i)17-s + (−0.446 − 4.33i)19-s + (−3.34 + 2.80i)21-s + (−0.0427 + 0.242i)23-s + (−0.939 + 0.342i)25-s + (−0.500 − 0.866i)27-s + (0.625 + 0.525i)29-s + (2.01 − 3.49i)31-s + ⋯
L(s)  = 1  + (−0.542 − 0.197i)3-s + (0.0776 + 0.440i)5-s + (0.825 − 1.42i)7-s + (0.255 + 0.214i)9-s + (−0.440 − 0.762i)11-s + (0.547 − 0.199i)13-s + (0.0448 − 0.254i)15-s + (−1.02 + 0.861i)17-s + (−0.102 − 0.994i)19-s + (−0.729 + 0.612i)21-s + (−0.00891 + 0.0505i)23-s + (−0.187 + 0.0684i)25-s + (−0.0962 − 0.166i)27-s + (0.116 + 0.0974i)29-s + (0.361 − 0.626i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(9.10294\)
Root analytic conductor: \(3.01710\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.156130647\)
\(L(\frac12)\) \(\approx\) \(1.156130647\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.446 + 4.33i)T \)
good7 \( 1 + (-2.18 + 3.78i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.46 + 2.53i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.97 + 0.718i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (4.23 - 3.55i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.0427 - 0.242i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.625 - 0.525i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-2.01 + 3.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.56T + 37T^{2} \)
41 \( 1 + (4.20 + 1.53i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.07 - 6.08i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (4.38 + 3.68i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-2.13 + 12.1i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-3.22 + 2.70i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.03 + 11.5i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.59 + 5.53i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.186 + 1.06i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-9.27 - 3.37i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-2.05 - 0.747i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.598 + 1.03i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.52 - 0.918i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (8.49 - 7.13i)T + (16.8 - 95.5i)T^{2} \)
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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

−9.812526879070436121794239211711, −8.483722119186183131639041846374, −7.959099925682916713212507946005, −6.92103803581523670875419880504, −6.41081219204901031026353910353, −5.25173210220969529826341995270, −4.38008533491222099084208026262, −3.42769430010356258043404168086, −1.90845904287810239283713466936, −0.54799718555404038782185779035, 1.58646227303429537130644354151, 2.60963656030058623360379987488, 4.22360387313932547683397419970, 5.01981040670941781840983403954, 5.63134581063760568895074530152, 6.56492844184211461930992424828, 7.65341123292392638685619937322, 8.620011319352878555457065414657, 9.069214874819335002154510459599, 10.07614089294449082103598558759

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