Properties

Label 2-1323-63.16-c1-0-11
Degree $2$
Conductor $1323$
Sign $-0.893 - 0.449i$
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 − 0.146·5-s − 0.446·8-s + (−0.150 − 0.260i)10-s − 1.66·11-s + (−0.0999 − 0.173i)13-s + (1.75 + 3.04i)16-s + (3.13 + 5.43i)17-s + (−3.45 + 5.99i)19-s + (0.162 − 0.280i)20-s + (−1.70 − 2.95i)22-s + 6.18·23-s − 4.97·25-s + (0.205 − 0.355i)26-s + ⋯
L(s)  = 1  + (0.726 + 1.25i)2-s + (−0.554 + 0.960i)4-s − 0.0654·5-s − 0.157·8-s + (−0.0474 − 0.0822i)10-s − 0.501·11-s + (−0.0277 − 0.0480i)13-s + (0.439 + 0.761i)16-s + (0.760 + 1.31i)17-s + (−0.793 + 1.37i)19-s + (0.0362 − 0.0627i)20-s + (−0.364 − 0.630i)22-s + 1.28·23-s − 0.995·25-s + (0.0402 − 0.0697i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.893 - 0.449i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.893 - 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.228910811\)
\(L(\frac12)\) \(\approx\) \(2.228910811\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.146T + 5T^{2} \)
11 \( 1 + 1.66T + 11T^{2} \)
13 \( 1 + (0.0999 + 0.173i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.13 - 5.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.45 - 5.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.18T + 23T^{2} \)
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 \( 1 + (3.50 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
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 \( 1 + (-2.67 - 4.62i)T + (-26.5 + 45.8i)T^{2} \)
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 \( 1 + (-0.778 - 1.34i)T + (-36.5 + 63.2i)T^{2} \)
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 \( 1 + (-4.53 + 7.85i)T + (-44.5 - 77.0i)T^{2} \)
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

−10.14127136430325395478338721729, −8.780210904012332709238677659699, −8.022570012109731511506217040863, −7.57351907355223970716549064128, −6.43541941143878906997211473662, −5.95449143721090686740368407306, −5.10348980165114368427630979712, −4.19326832008263646997867208591, −3.32811648214157249347660564913, −1.67877012269408684001893183902, 0.74056972453790423242065736501, 2.23788298637031179939629658456, 2.96003217703926217581328118601, 3.93061654062352131403673293181, 4.94203718574208915285190088960, 5.43362389473191061171253079556, 6.90770086190929098580227621257, 7.53143072058734623861675774655, 8.744406650918856873207685790538, 9.512913621031152551260563369492

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