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Properties

Label 2-1323-63.4-c1-0-8
Degree $2$
Conductor $1323$
Sign $-0.618 - 0.785i$
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  + (−0.119 + 0.207i)2-s + (0.971 + 1.68i)4-s − 1.18·5-s − 0.942·8-s + (0.141 − 0.244i)10-s + 3.70·11-s + (−0.5 + 0.866i)13-s + (−1.83 + 3.16i)16-s + (−3.47 + 6.01i)17-s + (−0.971 − 1.68i)19-s + (−1.14 − 1.98i)20-s + (−0.442 + 0.766i)22-s + 5.60·23-s − 3.60·25-s + (−0.119 − 0.207i)26-s + ⋯
L(s)  = 1  + (−0.0845 + 0.146i)2-s + (0.485 + 0.841i)4-s − 0.528·5-s − 0.333·8-s + (0.0446 − 0.0774i)10-s + 1.11·11-s + (−0.138 + 0.240i)13-s + (−0.457 + 0.792i)16-s + (−0.841 + 1.45i)17-s + (−0.222 − 0.385i)19-s + (−0.256 − 0.444i)20-s + (−0.0944 + 0.163i)22-s + 1.16·23-s − 0.720·25-s + (−0.0234 − 0.0406i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.618 - 0.785i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.618 - 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.278331976\)
\(L(\frac12)\) \(\approx\) \(1.278331976\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.119 - 0.207i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.18T + 5T^{2} \)
11 \( 1 - 3.70T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.47 - 6.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.971 + 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.60T + 23T^{2} \)
29 \( 1 + (-0.119 - 0.207i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.830 + 1.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.77 - 8.26i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.09 - 8.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.91 + 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.80 - 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.30 - 2.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.80 + 6.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.75 + 3.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.60T + 71T^{2} \)
73 \( 1 + (7.57 - 13.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.68 - 6.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.47 + 6.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.37 - 2.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.58 + 6.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.822153544572628874009402766575, −8.815072531247975747255935256241, −8.393594107028497016500190240333, −7.44948371086576249080242113247, −6.68910363248200799817580512610, −6.11669821013813795220867546525, −4.54450833531746815685850715548, −3.90620740469984902762392093388, −2.93286689312423162309713108111, −1.63683742336327449541235780238, 0.53497908559497811774259293031, 1.88031270194602658412712590091, 3.03647093777611315681052334090, 4.20231391995656008306432326709, 5.13647773957980000681438287648, 6.07882637713007986705473500060, 6.94726431851455812769583966929, 7.47715277318757677525918686895, 8.821795114270760571521447255191, 9.300051819339868363115019708370

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