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Properties

Label 2-1152-72.13-c1-0-24
Degree $2$
Conductor $1152$
Sign $0.941 - 0.337i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
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  + (−0.524 + 1.65i)3-s + (2.98 + 1.72i)5-s + (−2.02 − 3.50i)7-s + (−2.44 − 1.73i)9-s + (3.50 − 2.02i)11-s + (−0.866 − 0.5i)13-s + (−4.41 + 4.02i)15-s + 6.44·17-s + (6.84 − 1.5i)21-s + (1.11 − 1.92i)23-s + (3.44 + 5.97i)25-s + (4.14 − 3.13i)27-s + (5.10 − 2.94i)29-s + (2.93 − 5.07i)31-s + (1.5 + 6.84i)33-s + ⋯
L(s)  = 1  + (−0.302 + 0.953i)3-s + (1.33 + 0.771i)5-s + (−0.764 − 1.32i)7-s + (−0.816 − 0.577i)9-s + (1.05 − 0.609i)11-s + (−0.240 − 0.138i)13-s + (−1.13 + 1.03i)15-s + 1.56·17-s + (1.49 − 0.327i)21-s + (0.232 − 0.401i)23-s + (0.689 + 1.19i)25-s + (0.797 − 0.603i)27-s + (0.948 − 0.547i)29-s + (0.526 − 0.911i)31-s + (0.261 + 1.19i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.941 - 0.337i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.941 - 0.337i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.827817835\)
\(L(\frac12)\) \(\approx\) \(1.827817835\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.524 - 1.65i)T \)
good5 \( 1 + (-2.98 - 1.72i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.02 + 3.50i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.50 + 2.02i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-1.11 + 1.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.10 + 2.94i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.93 + 5.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.55iT - 37T^{2} \)
41 \( 1 + (4.17 - 7.22i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.93 - 5.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.02 - 3.50i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.34iT - 53T^{2} \)
59 \( 1 + (-1.92 - 1.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.64 + 0.949i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.6 - 7.88i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 - 3.55T + 73T^{2} \)
79 \( 1 + (4.74 + 8.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.92 - 1.11i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.44T + 89T^{2} \)
97 \( 1 + (8.17 + 14.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.912350679732735061279379457008, −9.551281747959775863715731982356, −8.338478460767812995344689784256, −7.08170956812166579585889135330, −6.29107999512587855992949498298, −5.85075983072992518117167297816, −4.59518875605273469105053894363, −3.56602638989356958953151598349, −2.88740666296962777817523165171, −0.997957393374914696165409342812, 1.26693940551400548998555163564, 2.10008244766868047171804561753, 3.22183489975502611513687668188, 5.13688519118432274588317391406, 5.48190081836642412833970315576, 6.42355932938554545769406341909, 6.93161979411040004816421393445, 8.339820505566200209087805765138, 8.927672504364467234416099829507, 9.674327377153610221351259989294

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