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Properties

Label 2-1323-63.25-c1-0-3
Degree $2$
Conductor $1323$
Sign $-0.703 - 0.711i$
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  − 0.670·2-s − 1.55·4-s + (−0.712 + 1.23i)5-s + 2.38·8-s + (0.477 − 0.827i)10-s + (−2.46 − 4.27i)11-s + (1.37 + 2.38i)13-s + 1.50·16-s + (0.559 − 0.969i)17-s + (2.00 + 3.47i)19-s + (1.10 − 1.91i)20-s + (1.65 + 2.86i)22-s + (2.71 − 4.70i)23-s + (1.48 + 2.57i)25-s + (−0.923 − 1.59i)26-s + ⋯
L(s)  = 1  − 0.473·2-s − 0.775·4-s + (−0.318 + 0.551i)5-s + 0.841·8-s + (0.151 − 0.261i)10-s + (−0.743 − 1.28i)11-s + (0.381 + 0.661i)13-s + 0.376·16-s + (0.135 − 0.235i)17-s + (0.460 + 0.797i)19-s + (0.247 − 0.427i)20-s + (0.352 + 0.610i)22-s + (0.566 − 0.981i)23-s + (0.296 + 0.514i)25-s + (−0.181 − 0.313i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.703 - 0.711i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.703 - 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4260149444\)
\(L(\frac12)\) \(\approx\) \(0.4260149444\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.670T + 2T^{2} \)
5 \( 1 + (0.712 - 1.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.46 + 4.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.37 - 2.38i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.559 + 0.969i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.00 - 3.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.71 + 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.40 - 5.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.50T + 31T^{2} \)
37 \( 1 + (-0.709 - 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.124 - 0.215i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.498 - 0.863i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.47T + 47T^{2} \)
53 \( 1 + (-0.410 + 0.710i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.58T + 59T^{2} \)
61 \( 1 + 0.0752T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + 0.0804T + 71T^{2} \)
73 \( 1 + (5.34 - 9.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 + (7.23 - 12.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.76 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.70 - 4.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.855026110699888530405367452531, −9.026597991910496189643622770024, −8.405358313918884171910345858901, −7.68209673046353272611347637171, −6.81096157573518790129136001415, −5.71866625690005212665657410139, −4.91578174256287537748040823768, −3.75673115570171331355476293732, −3.01139614429900924605184310798, −1.28641097730475574372882863875, 0.23956791994584263408497247871, 1.63187891132152316254784359013, 3.16409609697117601197940798274, 4.37912342705926221382120839713, 4.93565499795251242310140856933, 5.82584337572000479141232726822, 7.31079288103065972813610652947, 7.73371884616265091431299103727, 8.578298131436463693575028774695, 9.326933627045154904360722795316

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