Properties

Label 2-1323-9.7-c1-0-29
Degree $2$
Conductor $1323$
Sign $-0.759 + 0.650i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
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  + (1.02 − 1.77i)2-s + (−1.10 − 1.92i)4-s + (0.0731 + 0.126i)5-s − 0.446·8-s + 0.300·10-s + (0.832 − 1.44i)11-s + (−0.0999 − 0.173i)13-s + (1.75 − 3.04i)16-s − 6.27·17-s + 6.91·19-s + (0.162 − 0.280i)20-s + (−1.70 − 2.95i)22-s + (−3.09 − 5.35i)23-s + (2.48 − 4.31i)25-s − 0.410·26-s + ⋯
L(s)  = 1  + (0.726 − 1.25i)2-s + (−0.554 − 0.960i)4-s + (0.0327 + 0.0566i)5-s − 0.157·8-s + 0.0949·10-s + (0.250 − 0.434i)11-s + (−0.0277 − 0.0480i)13-s + (0.439 − 0.761i)16-s − 1.52·17-s + 1.58·19-s + (0.0362 − 0.0627i)20-s + (−0.364 − 0.630i)22-s + (−0.644 − 1.11i)23-s + (0.497 − 0.862i)25-s − 0.0805·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.759 + 0.650i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.759 + 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.439508416\)
\(L(\frac12)\) \(\approx\) \(2.439508416\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-1.02 + 1.77i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.0731 - 0.126i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.832 + 1.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0999 + 0.173i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.27T + 17T^{2} \)
19 \( 1 - 6.91T + 19T^{2} \)
23 \( 1 + (3.09 + 5.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.46 + 4.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.25 + 2.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.00T + 37T^{2} \)
41 \( 1 + (-1.15 - 2.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.940 - 1.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.905 + 1.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.34T + 53T^{2} \)
59 \( 1 + (-2.28 - 3.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.339 - 0.587i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.09 - 5.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.27T + 71T^{2} \)
73 \( 1 + 1.55T + 73T^{2} \)
79 \( 1 + (6.39 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.75 + 6.50i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.06T + 89T^{2} \)
97 \( 1 + (-3.98 + 6.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.649225116508854155495031869855, −8.673834308388774108844749676059, −7.76676097480972205246000622255, −6.67748257476639266662481231199, −5.79286838012200993782322017159, −4.66279936073337718705213456417, −4.11897990422893947793508526681, −2.96458119885490105476969578559, −2.25908392080953041463516217661, −0.840480271351090457548777352753, 1.61224841538661369189327699216, 3.22133610981040340881988187982, 4.25929566402359494103446553701, 5.04624737004680240330628119555, 5.75705299999127744001520410960, 6.72828442960874193523346789938, 7.25143153226255464362947823590, 8.014543496475980492202089066748, 9.024506367431147207329435683507, 9.678535637830480298987555321787

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