Properties

Label 2-1152-72.13-c1-0-8
Degree $2$
Conductor $1152$
Sign $0.301 - 0.953i$
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  + (−1.68 − 0.412i)3-s + (−0.379 − 0.219i)5-s + (2.14 + 3.71i)7-s + (2.65 + 1.38i)9-s + (3.41 − 1.97i)11-s + (−3.44 − 1.98i)13-s + (0.548 + 0.525i)15-s − 1.25·17-s + 6.44i·19-s + (−2.07 − 7.13i)21-s + (−2.32 + 4.02i)23-s + (−2.40 − 4.16i)25-s + (−3.90 − 3.43i)27-s + (5.46 − 3.15i)29-s + (−0.910 + 1.57i)31-s + ⋯
L(s)  = 1  + (−0.971 − 0.238i)3-s + (−0.169 − 0.0980i)5-s + (0.810 + 1.40i)7-s + (0.886 + 0.462i)9-s + (1.03 − 0.594i)11-s + (−0.954 − 0.551i)13-s + (0.141 + 0.135i)15-s − 0.304·17-s + 1.47i·19-s + (−0.452 − 1.55i)21-s + (−0.484 + 0.839i)23-s + (−0.480 − 0.832i)25-s + (−0.750 − 0.660i)27-s + (1.01 − 0.585i)29-s + (−0.163 + 0.283i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.301 - 0.953i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.070379093\)
\(L(\frac12)\) \(\approx\) \(1.070379093\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.412i)T \)
good5 \( 1 + (0.379 + 0.219i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.14 - 3.71i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.41 + 1.97i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.44 + 1.98i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.25T + 17T^{2} \)
19 \( 1 - 6.44iT - 19T^{2} \)
23 \( 1 + (2.32 - 4.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.46 + 3.15i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.910 - 1.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.87iT - 37T^{2} \)
41 \( 1 + (-1.53 + 2.65i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.45 + 0.838i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.01 - 8.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + (-1.80 - 1.04i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.8 - 6.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.98 + 1.14i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.21T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + (-5.15 - 8.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.81 - 5.09i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.57T + 89T^{2} \)
97 \( 1 + (3.78 + 6.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.02436450733878529557769440967, −9.158428070384393999380043464252, −8.179933390045502680689220606732, −7.62924606179578463277264950084, −6.27385768660992813662851793433, −5.84911459006678904418580279633, −4.98444492537123917972255530419, −4.04505429796918575837607237310, −2.47874273151619365781758202356, −1.31483162632247106154921173124, 0.58676946134562279387763331956, 1.92600442881147344274746793631, 3.82639545079980994967298633257, 4.51715203559823512619625614117, 5.05235632638176875064068059315, 6.57297277695287108486808830695, 6.98253702718242142139013988251, 7.68823204364584358006443239839, 9.025500126487744325497458108556, 9.762011520748112006844982047224

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