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Properties

Label 2-1323-63.16-c1-0-33
Degree $2$
Conductor $1323$
Sign $-0.296 + 0.954i$
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  + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s − 5-s + 3·8-s + (−0.5 − 0.866i)10-s − 5·11-s + (−2.5 − 4.33i)13-s + (0.500 + 0.866i)16-s + (−1.5 − 2.59i)17-s + (0.5 − 0.866i)19-s + (−0.500 + 0.866i)20-s + (−2.5 − 4.33i)22-s − 3·23-s − 4·25-s + (2.5 − 4.33i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s − 0.447·5-s + 1.06·8-s + (−0.158 − 0.273i)10-s − 1.50·11-s + (−0.693 − 1.20i)13-s + (0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + (0.114 − 0.198i)19-s + (−0.111 + 0.193i)20-s + (−0.533 − 0.923i)22-s − 0.625·23-s − 0.800·25-s + (0.490 − 0.849i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.296 + 0.954i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.296 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9498928835\)
\(L(\frac12)\) \(\approx\) \(0.9498928835\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + T + 5T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.5 + 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.566058338771663774021175690797, −8.086348578748196574546208856336, −7.79423267258019069834091983942, −6.98336899778822456885614904066, −5.94838930988178074035359883457, −5.20490723814767860951649364375, −4.62129109571402038655602621534, −3.19136169758370000246267150629, −2.15312717684805437439347255666, −0.31409392486082828486863177249, 1.91758619685677832436011503217, 2.67449525611020922780945133742, 3.86162443369783560225439791103, 4.47613313249703904211666298643, 5.53314221970534617385585153080, 6.70681787520667898924656578083, 7.64713445959511679428753362097, 7.984976831339952332742512854429, 9.111269242153439718006318770987, 10.10975908493741980216243300731

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