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Properties

Label 2-1176-7.2-c1-0-0
Degree $2$
Conductor $1176$
Sign $-0.749 + 0.661i$
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  + (0.5 + 0.866i)3-s + (−1.70 + 2.95i)5-s + (−0.499 + 0.866i)9-s + (−0.414 − 0.717i)11-s − 4.24·13-s − 3.41·15-s + (−3.70 − 6.42i)17-s + (−3.41 + 5.91i)19-s + (2.41 − 4.18i)23-s + (−3.32 − 5.76i)25-s − 0.999·27-s + 2.82·29-s + (1.41 + 2.44i)31-s + (0.414 − 0.717i)33-s + (0.828 − 1.43i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.763 + 1.32i)5-s + (−0.166 + 0.288i)9-s + (−0.124 − 0.216i)11-s − 1.17·13-s − 0.881·15-s + (−0.899 − 1.55i)17-s + (−0.783 + 1.35i)19-s + (0.503 − 0.871i)23-s + (−0.665 − 1.15i)25-s − 0.192·27-s + 0.525·29-s + (0.254 + 0.439i)31-s + (0.0721 − 0.124i)33-s + (0.136 − 0.235i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.749 + 0.661i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2425530881\)
\(L(\frac12)\) \(\approx\) \(0.2425530881\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (1.70 - 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.414 + 0.717i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + (3.70 + 6.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.41 - 5.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.41 + 4.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (-1.41 - 2.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.828 + 1.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + (-2.24 + 3.88i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.24 + 7.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.53 - 9.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.65 + 9.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + (3.87 + 6.71i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.82 - 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (2.87 - 4.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.242T + 97T^{2} \)
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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

−10.40897950143917754272701755519, −9.560988391829320781087929637952, −8.624801526350862192453876562586, −7.71917801825459096857699989134, −7.07396833367428870223029394783, −6.28530035786430797223743462655, −4.94630944632226708174643445030, −4.17022548377157239787902181174, −3.05284246044623820653025464842, −2.44955167162224465407072878229, 0.096669579733851806959884073859, 1.56655840831453853250316687495, 2.79907694696354914279291197333, 4.29078179741364707521276867065, 4.67482098570808980575486213072, 5.88433798386251881974248718681, 6.97390761611863246224838246919, 7.70111846801371911632411133226, 8.544849305818352039629889262456, 8.963097834016151816545897020221

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