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Properties

Label 2-1176-7.4-c1-0-18
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 + (−0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (−1.5 + 2.59i)11-s − 4·13-s − 0.999·15-s + (−2 − 3.46i)19-s + (−4 − 6.92i)23-s + (2 − 3.46i)25-s − 0.999·27-s − 3·29-s + (−2.5 + 4.33i)31-s + (1.5 + 2.59i)33-s + (−4 − 6.92i)37-s + (−2 + 3.46i)39-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.166 − 0.288i)9-s + (−0.452 + 0.783i)11-s − 1.10·13-s − 0.258·15-s + (−0.458 − 0.794i)19-s + (−0.834 − 1.44i)23-s + (0.400 − 0.692i)25-s − 0.192·27-s − 0.557·29-s + (−0.449 + 0.777i)31-s + (0.261 + 0.452i)33-s + (−0.657 − 1.13i)37-s + (−0.320 + 0.554i)39-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

\(L(1)\) \(\approx\) \(0.5707473149\)
\(L(\frac12)\) \(\approx\) \(0.5707473149\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (-5 - 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7T + 83T^{2} \)
89 \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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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

−9.221896871445982344606302617137, −8.587069911167798286202816501688, −7.63210037556117839773482525625, −7.09843605574030995706815060634, −6.11348381261918118207068011169, −4.92146307206248120599559016505, −4.30124119034943422008497122490, −2.81596252233036626975001297059, −1.97285711791520985186798182518, −0.21726418110364784231929934538, 1.97319637116671645053221329688, 3.18454238418686780597341920605, 3.88446471181808837278549144498, 5.11911815814601683872115214767, 5.79261903810868081611377339783, 7.00510921595633088003778287705, 7.77907743947412081275116844027, 8.474481515614371014088021203809, 9.505170649448269102904729746493, 10.06749897276807562380995735197

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