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Properties

Label 2-1323-9.4-c1-0-14
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: $\chi_{1323} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.759 - 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.467096748\)
\(L(\frac12)\) \(\approx\) \(2.467096748\)
\(L(\frac{3}{2})\) 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

−9.891892452656363496136280936314, −8.896396454199797510538666656739, −7.954168495121866801969189736103, −7.44532037915197968279730564655, −6.53276508974641129584064539005, −5.87491565942475849825983633760, −5.04161718175294764098386360835, −4.19488250136123352752075208605, −3.26323841605378477490503095117, −1.60664027728933336058966000271, 0.860730214906483060721373119715, 2.18457215755715969877600025682, 3.05106332261002786827055955494, 4.07557320469164908762789298992, 4.67155423005361601822363383079, 5.82535087658663935778993517796, 6.59889516463496301724560464753, 7.927342472390438604740184654941, 8.505850687458511856298092235634, 9.733680286567580142990932358613

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