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Properties

Label 2-1176-21.17-c1-0-28
Degree $2$
Conductor $1176$
Sign $-0.746 + 0.665i$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
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  + (−1.02 − 1.39i)3-s + (−0.244 − 0.423i)5-s + (−0.904 + 2.86i)9-s + (1.31 + 0.761i)11-s − 0.652i·13-s + (−0.341 + 0.775i)15-s + (3.77 − 6.54i)17-s + (0.364 − 0.210i)19-s + (−5.11 + 2.95i)23-s + (2.38 − 4.12i)25-s + (4.92 − 1.66i)27-s − 7.16i·29-s + (−6.39 − 3.69i)31-s + (−0.286 − 2.62i)33-s + (4.04 + 7.00i)37-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)3-s + (−0.109 − 0.189i)5-s + (−0.301 + 0.953i)9-s + (0.397 + 0.229i)11-s − 0.180i·13-s + (−0.0881 + 0.200i)15-s + (0.916 − 1.58i)17-s + (0.0836 − 0.0482i)19-s + (−1.06 + 0.616i)23-s + (0.476 − 0.824i)25-s + (0.947 − 0.320i)27-s − 1.33i·29-s + (−1.14 − 0.662i)31-s + (−0.0498 − 0.456i)33-s + (0.665 + 1.15i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.746 + 0.665i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.746 + 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9142652921\)
\(L(\frac12)\) \(\approx\) \(0.9142652921\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 + 1.39i)T \)
7 \( 1 \)
good5 \( 1 + (0.244 + 0.423i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.31 - 0.761i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.652iT - 13T^{2} \)
17 \( 1 + (-3.77 + 6.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.364 + 0.210i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.11 - 2.95i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.16iT - 29T^{2} \)
31 \( 1 + (6.39 + 3.69i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.04 - 7.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.53T + 41T^{2} \)
43 \( 1 + 5.56T + 43T^{2} \)
47 \( 1 + (4.65 + 8.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.26 + 0.731i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.67 + 6.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.65 - 5.57i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.31 - 12.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.8iT - 71T^{2} \)
73 \( 1 + (-8.00 - 4.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.607 + 1.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.03T + 83T^{2} \)
89 \( 1 + (4.77 + 8.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.95iT - 97T^{2} \)
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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

−9.634726432611621213529270578886, −8.423715366853462671940624033007, −7.71643146447744692309866764238, −7.02028361401949706847650189202, −6.09809568447721236433593810332, −5.32016002597123252351415979022, −4.39854643896850132428579868633, −3.02175218455236110502640464610, −1.79047128028968487160298493196, −0.44292954919824002636692485881, 1.48997388274708261618208640141, 3.29785748944395591755135057810, 3.90551448153218601661989377158, 4.99465862450517893487424728759, 5.87060420150621987584094411771, 6.53795486884365111332551776250, 7.62824205980084766227315123141, 8.640519076375596450149843266999, 9.314120647431845621061276636893, 10.25708256349848728907225582076

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