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Properties

Label 2-1176-7.4-c1-0-17
Degree $2$
Conductor $1176$
Sign $-0.991 + 0.126i$
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 + (−2.13 − 3.70i)5-s + (−0.499 − 0.866i)9-s + (2.13 − 3.70i)11-s + 1.27·13-s + 4.27·15-s + (−2 + 3.46i)17-s + (0.637 + 1.10i)19-s + (−2 − 3.46i)23-s + (−6.63 + 11.4i)25-s + 0.999·27-s − 2.27·29-s + (−0.5 + 0.866i)31-s + (2.13 + 3.70i)33-s + (−2.63 − 4.56i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.955 − 1.65i)5-s + (−0.166 − 0.288i)9-s + (0.644 − 1.11i)11-s + 0.353·13-s + 1.10·15-s + (−0.485 + 0.840i)17-s + (0.146 + 0.253i)19-s + (−0.417 − 0.722i)23-s + (−1.32 + 2.29i)25-s + 0.192·27-s − 0.422·29-s + (−0.0898 + 0.155i)31-s + (0.372 + 0.644i)33-s + (−0.433 − 0.751i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4340189706\)
\(L(\frac12)\) \(\approx\) \(0.4340189706\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 + 3.70i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.13 + 3.70i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.27T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.637 - 1.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.27T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.63 + 4.56i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 + 7.27T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.862 - 1.49i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.13 + 5.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.63 - 6.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-1.63 + 2.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.77 - 3.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.274T + 83T^{2} \)
89 \( 1 + (-2.27 - 3.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−9.146220317886029980273795025700, −8.501463631928614848634106966701, −8.210651989165400072063163773606, −6.81695554299455741788409266176, −5.77744173490622074560342659634, −5.04420104956756392718015910702, −4.04417137946267281504634604849, −3.58890387015977182604098150945, −1.47958167855104934382636022283, −0.19940388449532264378368721982, 1.87961689359995519293193694671, 3.05193540861414762761767418945, 3.91720846801189983315946156300, 5.01946102642159717060802657711, 6.42428770262686765925234944122, 6.85252584077591646100385419262, 7.47202026814764750484976998802, 8.261909323164355448427721447623, 9.515218098283392076910896896821, 10.24910161596364636037894392428

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