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Properties

Label 2-1176-21.17-c1-0-17
Degree $2$
Conductor $1176$
Sign $0.874 - 0.485i$
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  + (1.45 − 0.934i)3-s + (1.90 + 3.29i)5-s + (1.25 − 2.72i)9-s + (−0.309 − 0.178i)11-s + 4.04i·13-s + (5.84 + 3.02i)15-s + (0.0519 − 0.0900i)17-s + (2.12 − 1.22i)19-s + (1.15 − 0.665i)23-s + (−4.72 + 8.17i)25-s + (−0.723 − 5.14i)27-s − 4.97i·29-s + (6.83 + 3.94i)31-s + (−0.618 + 0.0287i)33-s + (5.45 + 9.45i)37-s + ⋯
L(s)  = 1  + (0.841 − 0.539i)3-s + (0.849 + 1.47i)5-s + (0.417 − 0.908i)9-s + (−0.0933 − 0.0538i)11-s + 1.12i·13-s + (1.50 + 0.780i)15-s + (0.0126 − 0.0218i)17-s + (0.487 − 0.281i)19-s + (0.240 − 0.138i)23-s + (−0.944 + 1.63i)25-s + (−0.139 − 0.990i)27-s − 0.923i·29-s + (1.22 + 0.708i)31-s + (−0.107 + 0.00501i)33-s + (0.896 + 1.55i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.874 - 0.485i$
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: \((2,\ 1176,\ (\ :1/2),\ 0.874 - 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.580477400\)
\(L(\frac12)\) \(\approx\) \(2.580477400\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.45 + 0.934i)T \)
7 \( 1 \)
good5 \( 1 + (-1.90 - 3.29i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.309 + 0.178i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.04iT - 13T^{2} \)
17 \( 1 + (-0.0519 + 0.0900i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.12 + 1.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.15 + 0.665i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.97iT - 29T^{2} \)
31 \( 1 + (-6.83 - 3.94i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.45 - 9.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.15T + 41T^{2} \)
43 \( 1 - 0.502T + 43T^{2} \)
47 \( 1 + (5.72 + 9.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.08 - 2.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.77 - 6.53i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.20 - 4.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.34 - 2.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.78iT - 71T^{2} \)
73 \( 1 + (-0.203 - 0.117i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.61 + 2.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.07T + 83T^{2} \)
89 \( 1 + (3.41 + 5.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.14iT - 97T^{2} \)
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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

−9.871851594043098563972278453086, −9.080904253660255718190667678873, −8.169243123318920563359162907799, −7.16758059459846263453881105842, −6.67135327937410268530521875294, −6.00532089313864837165522546995, −4.53388755554732744245125855080, −3.25402920233893803558181428499, −2.61299033428341586329270865827, −1.60727511152675931904396614561, 1.13449086820138527591697695384, 2.35507831211385125232536508160, 3.49537929006368003579024594266, 4.67855925529306185623926344072, 5.22801341279247606898452794698, 6.11425839970194214483182503643, 7.62247279432646692696299035217, 8.208336261397831618537860522055, 8.989972260978056691152504260745, 9.599454542308932081841779219877

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