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Properties

Label 2-1176-21.17-c1-0-3
Degree $2$
Conductor $1176$
Sign $0.0129 - 0.999i$
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  + (−1.72 − 0.187i)3-s + (0.244 + 0.423i)5-s + (2.92 + 0.647i)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 + (2.12 + 1.55i)33-s + (4.04 + 7.00i)37-s + ⋯
L(s)  = 1  + (−0.994 − 0.108i)3-s + (0.109 + 0.189i)5-s + (0.976 + 0.215i)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 + (0.370 + 0.271i)33-s + (0.665 + 1.15i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0129 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0129 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.0129 - 0.999i$
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: \((2,\ 1176,\ (\ :1/2),\ 0.0129 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8077464292\)
\(L(\frac12)\) \(\approx\) \(0.8077464292\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 + 0.187i)T \)
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

−10.35311776862093679586154510281, −9.158770863938326062565978284154, −8.342230003166284446975409670931, −7.32829112392746726225515946043, −6.52112280311240760468414758152, −5.87174364971434130798594012758, −4.90925627544329810823716945681, −4.05871903800960180364469375400, −2.66743666070217128985660674216, −1.26523029999947777795914494800, 0.43187570556132637984470797244, 1.97860007495552356819747356885, 3.41880532105097139313612209914, 4.72809202497078268113218820886, 5.14545443387144220576401710570, 6.16501478760256368755879035140, 7.10171598891693759505696025500, 7.61986599396818102736174709159, 9.202561292582024300355363101349, 9.361498505468593265614120372178

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