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Properties

Label 2-1323-63.4-c1-0-5
Degree $2$
Conductor $1323$
Sign $-0.967 + 0.252i$
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  + (−1.23 + 2.13i)2-s + (−2.02 − 3.51i)4-s + 2.59·5-s + 5.05·8-s + (−3.19 + 5.52i)10-s − 4.51·11-s + (−0.5 + 0.866i)13-s + (−2.16 + 3.74i)16-s + (−0.472 + 0.819i)17-s + (2.02 + 3.51i)19-s + (−5.25 − 9.10i)20-s + (5.55 − 9.61i)22-s + 0.273·23-s + 1.72·25-s + (−1.23 − 2.13i)26-s + ⋯
L(s)  = 1  + (−0.869 + 1.50i)2-s + (−1.01 − 1.75i)4-s + 1.15·5-s + 1.78·8-s + (−1.00 + 1.74i)10-s − 1.36·11-s + (−0.138 + 0.240i)13-s + (−0.540 + 0.936i)16-s + (−0.114 + 0.198i)17-s + (0.465 + 0.805i)19-s + (−1.17 − 2.03i)20-s + (1.18 − 2.05i)22-s + 0.0569·23-s + 0.345·25-s + (−0.241 − 0.417i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.967 + 0.252i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7298628577\)
\(L(\frac12)\) \(\approx\) \(0.7298628577\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 2.13i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.59T + 5T^{2} \)
11 \( 1 + 4.51T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.472 - 0.819i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.02 - 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.273T + 23T^{2} \)
29 \( 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.16 + 2.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.890 + 1.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.20 - 5.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.21 - 9.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.08 - 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.13 - 5.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.36 + 2.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.13 + 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.90 - 13.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 + (-0.753 + 1.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.35 - 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.472 + 0.819i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.17 + 12.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.74 - 9.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.792453882151679188418962778801, −9.255348137580683749413638740509, −8.228589446347102888491465106054, −7.75191349096464853731443439711, −6.82195327481966481281092210724, −5.97301978494337878250047827147, −5.51831059777579691564750979528, −4.62158125647881367813605239187, −2.79662716044767984207115529881, −1.42787887609306606369133999650, 0.40322055123403983971888824935, 1.88828143425112024942326679699, 2.54597394948905562684377700139, 3.45475130251187506488412073808, 4.88527795041514598952790044638, 5.63092439835807257128732629063, 6.94565904049979821272251518974, 7.924977700697490733089894260075, 8.738595614057078191547725517438, 9.425484555679529219652347656352

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