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Properties

Label 2-1170-65.49-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: $\chi_{1170} (829, \cdot )$
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

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

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