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Properties

Label 2-1323-63.5-c1-0-24
Degree $2$
Conductor $1323$
Sign $0.998 - 0.0549i$
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  + 2.27i·2-s − 3.18·4-s + (−0.717 − 1.24i)5-s − 2.69i·8-s + (2.82 − 1.63i)10-s + (2.80 + 1.61i)11-s + (−4.43 − 2.55i)13-s − 0.239·16-s + (−0.545 − 0.945i)17-s + (−3.88 − 2.24i)19-s + (2.28 + 3.95i)20-s + (−3.68 + 6.37i)22-s + (3.47 − 2.00i)23-s + (1.47 − 2.54i)25-s + (5.82 − 10.0i)26-s + ⋯
L(s)  = 1  + 1.60i·2-s − 1.59·4-s + (−0.320 − 0.555i)5-s − 0.951i·8-s + (0.894 − 0.516i)10-s + (0.844 + 0.487i)11-s + (−1.22 − 0.709i)13-s − 0.0597·16-s + (−0.132 − 0.229i)17-s + (−0.891 − 0.514i)19-s + (0.510 + 0.883i)20-s + (−0.784 + 1.35i)22-s + (0.723 − 0.417i)23-s + (0.294 − 0.509i)25-s + (1.14 − 1.97i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.998 - 0.0549i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.998 - 0.0549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8924012697\)
\(L(\frac12)\) \(\approx\) \(0.8924012697\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.27iT - 2T^{2} \)
5 \( 1 + (0.717 + 1.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.80 - 1.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.43 + 2.55i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.545 + 0.945i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.88 + 2.24i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.47 + 2.00i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.02 - 0.593i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.74iT - 31T^{2} \)
37 \( 1 + (-0.119 + 0.207i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.71 + 6.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.82 + 6.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.22T + 47T^{2} \)
53 \( 1 + (6.07 - 3.50i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.47T + 59T^{2} \)
61 \( 1 - 3.26iT - 61T^{2} \)
67 \( 1 - 0.660T + 67T^{2} \)
71 \( 1 + 3.82iT - 71T^{2} \)
73 \( 1 + (-6.33 + 3.65i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.66T + 79T^{2} \)
83 \( 1 + (5.45 + 9.44i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.84 - 11.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.69 - 1.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.188382461256485609697164578593, −8.720771831348387249345592730133, −7.84692102863688136165284738323, −7.13664771061898706238260959087, −6.54913618518678347749606513825, −5.48767713979588855020520202328, −4.77019782642841357610161249312, −4.09991577374496306127701049099, −2.41727705028555658134155831583, −0.39053233657494395826563996783, 1.33052571269499076179172397297, 2.42574406384415338963474397477, 3.35038358251662027100124619839, 4.12230651417445696804294786446, 5.00906141157984034299957210061, 6.42701103209617065052933052461, 7.12981750159550319647896541647, 8.304197200295437184364616211683, 9.192628851022260464546905931256, 9.744560055530089001602438396333

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