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Properties

Label 2-1152-9.7-c1-0-22
Degree $2$
Conductor $1152$
Sign $0.849 + 0.527i$
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  + (−0.970 − 1.43i)3-s + (1.07 + 1.86i)5-s + (0.153 − 0.265i)7-s + (−1.11 + 2.78i)9-s + (2.50 − 4.34i)11-s + (0.470 + 0.815i)13-s + (1.62 − 3.34i)15-s − 4.70·17-s + 1.61·19-s + (−0.529 + 0.0378i)21-s + (4.08 + 7.06i)23-s + (0.191 − 0.330i)25-s + (5.07 − 1.10i)27-s + (2.39 − 4.14i)29-s + (−1.29 − 2.24i)31-s + ⋯
L(s)  = 1  + (−0.560 − 0.828i)3-s + (0.480 + 0.832i)5-s + (0.0578 − 0.100i)7-s + (−0.371 + 0.928i)9-s + (0.755 − 1.30i)11-s + (0.130 + 0.226i)13-s + (0.419 − 0.864i)15-s − 1.14·17-s + 0.371·19-s + (−0.115 + 0.00825i)21-s + (0.851 + 1.47i)23-s + (0.0382 − 0.0661i)25-s + (0.977 − 0.212i)27-s + (0.444 − 0.770i)29-s + (−0.233 − 0.403i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.849 + 0.527i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.849 + 0.527i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.510614151\)
\(L(\frac12)\) \(\approx\) \(1.510614151\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.970 + 1.43i)T \)
good5 \( 1 + (-1.07 - 1.86i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.153 + 0.265i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.50 + 4.34i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.470 - 0.815i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.70T + 17T^{2} \)
19 \( 1 - 1.61T + 19T^{2} \)
23 \( 1 + (-4.08 - 7.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.39 + 4.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.29 + 2.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + (-3.86 - 6.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.138 + 0.239i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.92 + 3.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.23T + 53T^{2} \)
59 \( 1 + (4.95 + 8.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.36 + 9.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.02 - 3.50i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.59T + 71T^{2} \)
73 \( 1 + 5.43T + 73T^{2} \)
79 \( 1 + (-8.30 + 14.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.91 + 5.05i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.94T + 89T^{2} \)
97 \( 1 + (-7.07 + 12.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.719743678028509727152238203671, −8.906543892807022442762501230045, −7.919273507335555084903501912238, −7.09738788863482402452178374681, −6.23320756287361097272911860635, −5.94599622649257253454690063214, −4.63509811493294908190118405170, −3.30403453192159362953913258011, −2.26801090116287275569149377307, −0.942901503537792649888675241023, 1.05635430754212010391320755716, 2.54997365300528617175514615245, 4.10402336504170958981389524109, 4.67504299641266047415804689473, 5.43426266677747681536633834133, 6.44745907717633355937772941271, 7.18434742281993770958133784816, 8.723248507955932495163414445527, 9.039343718918982565777777595226, 9.809589939455251924017597044168

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