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Properties

Label 2-1176-7.4-c1-0-12
Degree $2$
Conductor $1176$
Sign $0.749 + 0.661i$
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 + (0.292 + 0.507i)5-s + (−0.499 − 0.866i)9-s + (2.41 − 4.18i)11-s − 4.24·13-s − 0.585·15-s + (2.29 − 3.97i)17-s + (0.585 + 1.01i)19-s + (−0.414 − 0.717i)23-s + (2.32 − 4.03i)25-s + 0.999·27-s − 2.82·29-s + (1.41 − 2.44i)31-s + (2.41 + 4.18i)33-s + (−4.82 − 8.36i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.130 + 0.226i)5-s + (−0.166 − 0.288i)9-s + (0.727 − 1.26i)11-s − 1.17·13-s − 0.151·15-s + (0.556 − 0.963i)17-s + (0.134 + 0.232i)19-s + (−0.0863 − 0.149i)23-s + (0.465 − 0.806i)25-s + 0.192·27-s − 0.525·29-s + (0.254 − 0.439i)31-s + (0.420 + 0.727i)33-s + (−0.793 − 1.37i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.749 + 0.661i$
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: \((2,\ 1176,\ (\ :1/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.319786505\)
\(L(\frac12)\) \(\approx\) \(1.319786505\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 0.507i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.41 + 4.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + (-2.29 + 3.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.585 - 1.01i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.414 + 0.717i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 + (-1.41 + 2.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.82 + 8.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.75T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + (-6.24 - 10.8i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.24 - 7.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.53 + 2.65i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.65 + 9.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.48T + 71T^{2} \)
73 \( 1 + (-8.12 + 14.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.17 + 2.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-7.12 - 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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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

−9.488998487972263543138534545164, −9.203018401390361772362897597897, −8.016595559755904173035369601219, −7.18773154113758360477390011620, −6.18699493247350672131402267781, −5.48545744766989326575180525085, −4.50121206514456436413080807471, −3.48587155625044422623133700374, −2.48850594356683993062454718487, −0.64395425259391609897804103835, 1.32448904632880503791278376326, 2.35013570077207106368376454926, 3.78093746369659012679676131947, 4.85239784459324301266961169845, 5.57069400233169615923179306955, 6.76289244480039667425718478094, 7.21586285260258071687928179267, 8.134826919329298407526005761705, 9.142451719339674553326746447430, 9.843941638702098996233778211774

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