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Properties

Label 2-1176-21.17-c1-0-16
Degree $2$
Conductor $1176$
Sign $0.996 - 0.0800i$
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  + (−1.71 + 0.247i)3-s + (1.28 + 2.23i)5-s + (2.87 − 0.849i)9-s + (1.43 + 0.826i)11-s − 5.71i·13-s + (−2.76 − 3.50i)15-s + (3.79 − 6.56i)17-s + (−2.58 + 1.49i)19-s + (0.249 − 0.143i)23-s + (−0.825 + 1.43i)25-s + (−4.72 + 2.16i)27-s + 2.05i·29-s + (5.21 + 3.00i)31-s + (−2.65 − 1.06i)33-s + (−0.877 − 1.51i)37-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)3-s + (0.576 + 0.998i)5-s + (0.959 − 0.283i)9-s + (0.431 + 0.249i)11-s − 1.58i·13-s + (−0.713 − 0.906i)15-s + (0.919 − 1.59i)17-s + (−0.594 + 0.343i)19-s + (0.0519 − 0.0300i)23-s + (−0.165 + 0.286i)25-s + (−0.908 + 0.417i)27-s + 0.382i·29-s + (0.936 + 0.540i)31-s + (−0.462 − 0.184i)33-s + (−0.144 − 0.249i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.996 - 0.0800i$
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: \((2,\ 1176,\ (\ :1/2),\ 0.996 - 0.0800i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.366461671\)
\(L(\frac12)\) \(\approx\) \(1.366461671\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 - 0.247i)T \)
7 \( 1 \)
good5 \( 1 + (-1.28 - 2.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.43 - 0.826i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.71iT - 13T^{2} \)
17 \( 1 + (-3.79 + 6.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.58 - 1.49i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.249 + 0.143i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.05iT - 29T^{2} \)
31 \( 1 + (-5.21 - 3.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.877 + 1.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.28T + 41T^{2} \)
43 \( 1 - 2.46T + 43T^{2} \)
47 \( 1 + (0.186 + 0.323i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.73 - 3.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.89 + 8.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.889 - 0.513i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.18 - 2.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + (-3.30 - 1.90i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.56 - 7.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.65T + 83T^{2} \)
89 \( 1 + (-7.25 - 12.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.43iT - 97T^{2} \)
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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

−10.08273642769668311404894337794, −9.267797288601785218163513242824, −7.915951950892947298588440785589, −7.12369833769234519981357359718, −6.39016085065651314725619230779, −5.60470696239927634516960600007, −4.87181580910212344270824061395, −3.53425243490957834990092606322, −2.54367280975286062005568915723, −0.874467150911573995404165328762, 1.08303094233174441607236722096, 1.98327676795176062405360476950, 3.99627098237960020000731742749, 4.59611209724695182873255039839, 5.69629845454640892129814724318, 6.20596757548200822793284914480, 7.07666049749654006584373827759, 8.246581143504843118838545777042, 9.004379344990147782620411774864, 9.795566942254776814665525284684

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