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Properties

Label 2-1323-63.59-c1-0-14
Degree $2$
Conductor $1323$
Sign $-0.457 + 0.889i$
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  + (−1.81 − 1.04i)2-s + (1.19 + 2.07i)4-s − 2.08·5-s − 0.819i·8-s + (3.79 + 2.18i)10-s + 3.22i·11-s + (−2.68 − 1.55i)13-s + (1.53 − 2.65i)16-s + (−0.816 + 1.41i)17-s + (−4.79 + 2.76i)19-s + (−2.49 − 4.32i)20-s + (3.38 − 5.85i)22-s + 1.16i·23-s − 0.632·25-s + (3.25 + 5.63i)26-s + ⋯
L(s)  = 1  + (−1.28 − 0.740i)2-s + (0.597 + 1.03i)4-s − 0.934·5-s − 0.289i·8-s + (1.19 + 0.692i)10-s + 0.973i·11-s + (−0.745 − 0.430i)13-s + (0.383 − 0.663i)16-s + (−0.197 + 0.342i)17-s + (−1.09 + 0.634i)19-s + (−0.558 − 0.967i)20-s + (0.721 − 1.24i)22-s + 0.242i·23-s − 0.126·25-s + (0.637 + 1.10i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.457 + 0.889i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (962, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.457 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3566294069\)
\(L(\frac12)\) \(\approx\) \(0.3566294069\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 + 1.04i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 2.08T + 5T^{2} \)
11 \( 1 - 3.22iT - 11T^{2} \)
13 \( 1 + (2.68 + 1.55i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.816 - 1.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.79 - 2.76i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.16iT - 23T^{2} \)
29 \( 1 + (-7.05 + 4.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.16 + 2.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.82 - 4.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.35 - 2.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.974 + 1.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.06 + 7.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.27 + 3.04i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.98 + 3.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.15 - 2.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.336 - 0.583i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.01iT - 71T^{2} \)
73 \( 1 + (-2.96 - 1.71i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.07 + 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.54 - 2.67i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.45 + 4.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.07 - 1.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.612042226007565003171479516857, −8.414698695084174514737459481312, −8.089289292773963472388991476951, −7.33104197422499509365447059375, −6.31626845423053851171183063901, −4.90206019049805742086995669246, −4.05022952809646493743676241348, −2.79621732303300811555587182833, −1.84265029927547234331769728025, −0.31848634840367176396919819807, 0.852470743162841737030884182186, 2.62267429505970560762524227748, 3.92084885720026691058056743791, 4.87665367702563653816339080895, 6.21700241181130510281362620741, 6.81950746487572666241373678035, 7.61940999576995889101725054237, 8.334231567025854959433258958789, 8.841784395836081647953736147537, 9.628165643551331768328669634686

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