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Properties

Label 2-1176-56.27-c1-0-61
Degree $2$
Conductor $1176$
Sign $-0.997 - 0.0752i$
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  + (−0.298 − 1.38i)2-s − i·3-s + (−1.82 + 0.825i)4-s − 0.310·5-s + (−1.38 + 0.298i)6-s + (1.68 + 2.27i)8-s − 9-s + (0.0927 + 0.429i)10-s + 1.24·11-s + (0.825 + 1.82i)12-s + 2.68·13-s + 0.310i·15-s + (2.63 − 3.00i)16-s − 2.22i·17-s + (0.298 + 1.38i)18-s − 5.93i·19-s + ⋯
L(s)  = 1  + (−0.211 − 0.977i)2-s − 0.577i·3-s + (−0.910 + 0.412i)4-s − 0.138·5-s + (−0.564 + 0.121i)6-s + (0.595 + 0.803i)8-s − 0.333·9-s + (0.0293 + 0.135i)10-s + 0.375·11-s + (0.238 + 0.525i)12-s + 0.745·13-s + 0.0801i·15-s + (0.659 − 0.752i)16-s − 0.540i·17-s + (0.0704 + 0.325i)18-s − 1.36i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.997 - 0.0752i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.997 - 0.0752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9503744257\)
\(L(\frac12)\) \(\approx\) \(0.9503744257\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.298 + 1.38i)T \)
3 \( 1 + iT \)
7 \( 1 \)
good5 \( 1 + 0.310T + 5T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 - 2.68T + 13T^{2} \)
17 \( 1 + 2.22iT - 17T^{2} \)
19 \( 1 + 5.93iT - 19T^{2} \)
23 \( 1 + 3.30iT - 23T^{2} \)
29 \( 1 + 0.191iT - 29T^{2} \)
31 \( 1 + 3.91T + 31T^{2} \)
37 \( 1 - 0.743iT - 37T^{2} \)
41 \( 1 + 9.28iT - 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 12.4iT - 53T^{2} \)
59 \( 1 + 5.96iT - 59T^{2} \)
61 \( 1 + 9.17T + 61T^{2} \)
67 \( 1 - 4.51T + 67T^{2} \)
71 \( 1 - 7.92iT - 71T^{2} \)
73 \( 1 + 8.05iT - 73T^{2} \)
79 \( 1 + 15.3iT - 79T^{2} \)
83 \( 1 + 14.2iT - 83T^{2} \)
89 \( 1 + 6.38iT - 89T^{2} \)
97 \( 1 - 1.62iT - 97T^{2} \)
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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

−9.152249693225207983686576303096, −8.862276058844950558379064551243, −7.81027789006275890199014867074, −7.04602556004743815563176791781, −5.94794721894636737076994459369, −4.86188348538608894876585823852, −3.87387825702828155042080475617, −2.85702561154697589983844139294, −1.78518995035506562727532926877, −0.46813527450833066405321073844, 1.51601084724420534431993541603, 3.58841046213743608616542958028, 4.09098328643506259013737348489, 5.33080923980583617611773492923, 5.96298049716609189359005607843, 6.81060797561473625646165863914, 7.942131515475095301288743962032, 8.369200081017556829177924677924, 9.366668785396882590699477606367, 9.927554478196413258892082564922

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