Properties

Label 2-1152-9.7-c1-0-28
Degree $2$
Conductor $1152$
Sign $0.470 + 0.882i$
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  + (0.564 + 1.63i)3-s + (−1.59 − 2.75i)5-s + (−0.607 + 1.05i)7-s + (−2.36 + 1.84i)9-s + (0.312 − 0.540i)11-s + (−1.06 − 1.84i)13-s + (3.61 − 4.16i)15-s − 1.83·17-s + 7.15·19-s + (−2.06 − 0.400i)21-s + (−0.780 − 1.35i)23-s + (−2.57 + 4.46i)25-s + (−4.36 − 2.82i)27-s + (4.87 − 8.44i)29-s + (−3.32 − 5.75i)31-s + ⋯
L(s)  = 1  + (0.325 + 0.945i)3-s + (−0.712 − 1.23i)5-s + (−0.229 + 0.397i)7-s + (−0.787 + 0.616i)9-s + (0.0941 − 0.163i)11-s + (−0.295 − 0.511i)13-s + (0.934 − 1.07i)15-s − 0.445·17-s + 1.64·19-s + (−0.450 − 0.0874i)21-s + (−0.162 − 0.282i)23-s + (−0.515 + 0.892i)25-s + (−0.839 − 0.543i)27-s + (0.905 − 1.56i)29-s + (−0.596 − 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.470 + 0.882i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.470 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.152457287\)
\(L(\frac12)\) \(\approx\) \(1.152457287\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.564 - 1.63i)T \)
good5 \( 1 + (1.59 + 2.75i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.607 - 1.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.312 + 0.540i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.06 + 1.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.83T + 17T^{2} \)
19 \( 1 - 7.15T + 19T^{2} \)
23 \( 1 + (0.780 + 1.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.87 + 8.44i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.32 + 5.75i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.73T + 37T^{2} \)
41 \( 1 + (5.64 + 9.77i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.51 + 7.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.36 + 2.35i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.60T + 53T^{2} \)
59 \( 1 + (4.02 + 6.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.79 - 4.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.95 + 6.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + 5.66T + 73T^{2} \)
79 \( 1 + (3.21 - 5.56i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.27 - 5.67i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.02T + 89T^{2} \)
97 \( 1 + (4.70 - 8.14i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.459893912360064360581053289673, −8.982297951350093782401280198245, −8.127121949672823363261820661162, −7.56097744753692853659814725591, −5.97030067887101398434648584649, −5.21386506269995335596231755365, −4.42436905312866721565455350375, −3.61687527274812721510907622005, −2.46049545582802936500279799415, −0.50992145680409116692848253304, 1.37794574017931649999971790709, 2.90916328301978180133269865419, 3.34363556815537126674721168329, 4.68452866995959789518460933244, 6.07037338158544352845814909678, 6.94721607528859592829320053545, 7.25773922458693395162484383271, 8.030957435571295896792138736161, 9.077765213830506037307198338388, 9.909965828568290903405400377018

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