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Properties

Label 2-1170-65.4-c1-0-21
Degree $2$
Conductor $1170$
Sign $0.839 + 0.543i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.21 − 0.339i)5-s + (2.39 + 4.15i)7-s − 0.999·8-s + (0.811 − 2.08i)10-s + (−1.43 − 0.825i)11-s + (−1.09 − 3.43i)13-s + 4.79·14-s + (−0.5 + 0.866i)16-s + (0.0697 − 0.0402i)17-s + (3.67 − 2.12i)19-s + (−1.39 − 1.74i)20-s + (−1.43 + 0.825i)22-s + (5.10 + 2.94i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.988 − 0.151i)5-s + (0.906 + 1.57i)7-s − 0.353·8-s + (0.256 − 0.658i)10-s + (−0.431 − 0.248i)11-s + (−0.302 − 0.953i)13-s + 1.28·14-s + (−0.125 + 0.216i)16-s + (0.0169 − 0.00976i)17-s + (0.843 − 0.487i)19-s + (−0.312 − 0.390i)20-s + (−0.304 + 0.176i)22-s + (1.06 + 0.615i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.839 + 0.543i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.839 + 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.506915365\)
\(L(\frac12)\) \(\approx\) \(2.506915365\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-2.21 + 0.339i)T \)
13 \( 1 + (1.09 + 3.43i)T \)
good7 \( 1 + (-2.39 - 4.15i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.43 + 0.825i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.0697 + 0.0402i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.67 + 2.12i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.10 - 2.94i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.21 - 3.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.06iT - 31T^{2} \)
37 \( 1 + (-0.908 + 1.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.87 - 3.39i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.35 + 4.24i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.448T + 47T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 + (1.82 - 1.05i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.56 - 2.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.36 + 3.67i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + (7.23 + 4.17i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.90 + 5.02i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.633525507707312537480931943260, −9.047263333302468367249282589149, −8.346449077195999045134383096100, −7.22648594788850289998719007997, −5.82821829965422296596641442936, −5.39659492269713555085234374875, −4.85891112633542080790700895930, −3.08141445069800115294748202460, −2.45727548819684809481300814570, −1.31876187897860238378166933302, 1.23354752730937458170488031603, 2.58125808042253930416452679600, 4.04937254032562250258342765428, 4.69783787639513697127707875856, 5.60387454175983863760132892320, 6.59697586685632539499662592088, 7.37171672098931143406715640564, 7.86256389888422550940015475396, 9.105103476035494045032585280992, 9.780844286833242403126033140358

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