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Properties

Label 2-1323-63.16-c1-0-14
Degree $2$
Conductor $1323$
Sign $-0.997 + 0.0674i$
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  + (−1.19 − 2.06i)2-s + (−1.84 + 3.20i)4-s − 2.92·5-s + 4.05·8-s + (3.48 + 6.03i)10-s + 1.35·11-s + (0.733 + 1.26i)13-s + (−1.13 − 1.96i)16-s + (1.65 + 2.86i)17-s + (1.10 − 1.91i)19-s + (5.39 − 9.35i)20-s + (−1.61 − 2.79i)22-s − 2.62·23-s + 3.53·25-s + (1.74 − 3.03i)26-s + ⋯
L(s)  = 1  + (−0.843 − 1.46i)2-s + (−0.924 + 1.60i)4-s − 1.30·5-s + 1.43·8-s + (1.10 + 1.90i)10-s + 0.408·11-s + (0.203 + 0.352i)13-s + (−0.284 − 0.492i)16-s + (0.401 + 0.695i)17-s + (0.253 − 0.438i)19-s + (1.20 − 2.09i)20-s + (−0.344 − 0.596i)22-s − 0.548·23-s + 0.706·25-s + (0.343 − 0.594i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.997 + 0.0674i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.997 + 0.0674i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4825244007\)
\(L(\frac12)\) \(\approx\) \(0.4825244007\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 2.06i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.92T + 5T^{2} \)
11 \( 1 - 1.35T + 11T^{2} \)
13 \( 1 + (-0.733 - 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.65 - 2.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.10 + 1.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.62T + 23T^{2} \)
29 \( 1 + (0.521 - 0.903i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.63 + 2.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.43 + 9.41i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.904 + 1.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.17 - 3.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.98 + 3.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.22 - 5.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.10 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.279 - 0.484i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.40 - 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (5.22 + 9.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.383 + 0.664i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.983 - 1.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.20 + 5.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.14 + 7.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.320704896004490219028488364700, −8.589996840756774052350721528809, −7.943809767788644015392521607041, −7.18548707774364265672435317836, −5.90500289442626579986568664418, −4.32861742693188400581514890494, −3.84778249562136170793697572474, −2.90868165802798821338704216680, −1.64947092608875500987255650353, −0.35361142095204724433957693583, 1.00448553112002162110624728045, 3.16188406493983021317666180037, 4.26382929158914167913569188511, 5.22197559563632189717480746330, 6.17283477903320150785475211445, 6.96900512724098688012010484894, 7.72488364282575550084749822585, 8.151659787141722870449335884760, 8.899693327800512358111724624059, 9.782626446500782347380772666306

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