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Properties

Label 2-1176-7.4-c1-0-1
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 + (1.63 + 2.83i)5-s + (−0.499 − 0.866i)9-s + (−1.63 + 2.83i)11-s − 6.27·13-s − 3.27·15-s + (−2 + 3.46i)17-s + (−3.13 − 5.43i)19-s + (−2 − 3.46i)23-s + (−2.86 + 4.95i)25-s + 0.999·27-s + 5.27·29-s + (−0.5 + 0.866i)31-s + (−1.63 − 2.83i)33-s + (1.13 + 1.97i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.732 + 1.26i)5-s + (−0.166 − 0.288i)9-s + (−0.493 + 0.855i)11-s − 1.74·13-s − 0.845·15-s + (−0.485 + 0.840i)17-s + (−0.719 − 1.24i)19-s + (−0.417 − 0.722i)23-s + (−0.572 + 0.991i)25-s + 0.192·27-s + 0.979·29-s + (−0.0898 + 0.155i)31-s + (−0.285 − 0.493i)33-s + (0.186 + 0.323i)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

\(L(1)\) \(\approx\) \(0.7197675007\)
\(L(\frac12)\) \(\approx\) \(0.7197675007\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 2.83i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.63 - 2.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.27T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.13 + 5.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.27T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.13 - 1.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.54T + 41T^{2} \)
43 \( 1 - 0.274T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.63 - 8.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.637 - 1.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.137 + 0.238i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (2.13 - 3.70i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.77 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.27T + 83T^{2} \)
89 \( 1 + (5.27 + 9.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.72T + 97T^{2} \)
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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

−10.29260265823705295926553396942, −9.687219011089317211353217780845, −8.719487919197769931347104700182, −7.49991701060940296001508941787, −6.80512260048271359462295050699, −6.14626560195690827268689998232, −4.98323915875044978100695559708, −4.32085149648720455609570160613, −2.74653797183982337296454501122, −2.27959762688732794083328690386, 0.29436549027709707823040348888, 1.70143797118374783547436750214, 2.73033687499624013240957497978, 4.37908555838206050748524471014, 5.22271490222358933032557084005, 5.77763686053225862967228062558, 6.79758672154317782143429289794, 7.86692022552947846224847675077, 8.412073796188218204706000382867, 9.487926867028089314624589336545

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