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Properties

Label 2-1323-63.16-c1-0-27
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: $\chi_{1323} (667, \cdot )$
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

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

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