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Properties

Label 2-1176-21.17-c1-0-26
Degree $2$
Conductor $1176$
Sign $0.961 + 0.273i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.961 + 0.273i$
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: \((2,\ 1176,\ (\ :1/2),\ 0.961 + 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.389305400\)
\(L(\frac12)\) \(\approx\) \(2.389305400\)
\(L(\frac{3}{2})\) 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

−9.838545842125713139222799600748, −8.604396539039007591133880415702, −8.458114726915175161679362606449, −7.25306011081375987440946162417, −6.72544259869085105422449191320, −5.18111092062769897748934013076, −4.62125990250697784518101016379, −3.26268660432764290840351973202, −2.68940768201947629550243064182, −1.11626584596565416387482128462, 1.36666476589109391918763218068, 2.61285824348217937893015717595, 3.51273330313323556791990606604, 4.41233347757333584180996543299, 5.61851904136630583340829079874, 6.60505846046064606103428023380, 7.63282230467917347459111609439, 8.030463859262314993052733411366, 8.954980431686247505573920166639, 9.789742750509651602464113888654

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