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Properties

Label 2-1323-63.4-c1-0-30
Degree $2$
Conductor $1323$
Sign $0.230 + 0.972i$
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  + (−0.551 + 0.955i)2-s + (0.391 + 0.678i)4-s + 0.105·5-s − 3.07·8-s + (−0.0581 + 0.100i)10-s − 3.33·11-s + (1.23 − 2.14i)13-s + (0.909 − 1.57i)16-s + (−0.806 + 1.39i)17-s + (−3.84 − 6.65i)19-s + (0.0413 + 0.0715i)20-s + (1.84 − 3.18i)22-s + 1.89·23-s − 4.98·25-s + (1.36 + 2.36i)26-s + ⋯
L(s)  = 1  + (−0.389 + 0.675i)2-s + (0.195 + 0.339i)4-s + 0.0471·5-s − 1.08·8-s + (−0.0183 + 0.0318i)10-s − 1.00·11-s + (0.343 − 0.595i)13-s + (0.227 − 0.393i)16-s + (−0.195 + 0.338i)17-s + (−0.881 − 1.52i)19-s + (0.00924 + 0.0160i)20-s + (0.392 − 0.679i)22-s + 0.395·23-s − 0.997·25-s + (0.268 + 0.464i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.230 + 0.972i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.230 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4059294604\)
\(L(\frac12)\) \(\approx\) \(0.4059294604\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.551 - 0.955i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 0.105T + 5T^{2} \)
11 \( 1 + 3.33T + 11T^{2} \)
13 \( 1 + (-1.23 + 2.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.806 - 1.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.84 + 6.65i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.89T + 23T^{2} \)
29 \( 1 + (4.64 + 8.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.63 - 8.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.991 - 1.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.74 + 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.77 + 6.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.59 + 2.76i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.98 - 8.64i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.22 + 3.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.83 - 4.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.98 + 8.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.29T + 71T^{2} \)
73 \( 1 + (2.36 - 4.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.84 - 6.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.584 + 1.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.01 + 5.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.90 - 3.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.166769604766946116525273398250, −8.536936562867710984985274764754, −7.82739614439402438205404261747, −7.13517730727528744873470233389, −6.25825835496212433565456999493, −5.51058842118102229809312279943, −4.38081692401445873985806769445, −3.16216830123756714300905253351, −2.28347172434467710445462405380, −0.17894846591443560373161402463, 1.49970761956413755700382368555, 2.41281920301729260379776530964, 3.49619909267232752581310938667, 4.66849284595575029733714466805, 5.80899535305898113114334656317, 6.31620740735829287757456847999, 7.52008118242225364408671489202, 8.300499423685473518504675073510, 9.232739773252226083821471697738, 9.884930184058735543590033805639

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