Properties

Label 2-1323-63.58-c1-0-31
Degree $2$
Conductor $1323$
Sign $-0.990 - 0.134i$
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  − 0.239·2-s − 1.94·4-s + (−1.29 − 2.24i)5-s + 0.942·8-s + (0.309 + 0.536i)10-s + (2.09 − 3.62i)11-s + (−1.84 + 3.18i)13-s + 3.66·16-s + (0.855 + 1.48i)17-s + (3.57 − 6.19i)19-s + (2.51 + 4.36i)20-s + (−0.499 + 0.866i)22-s + (−2.56 − 4.43i)23-s + (−0.858 + 1.48i)25-s + (0.440 − 0.762i)26-s + ⋯
L(s)  = 1  − 0.169·2-s − 0.971·4-s + (−0.579 − 1.00i)5-s + 0.333·8-s + (0.0979 + 0.169i)10-s + (0.630 − 1.09i)11-s + (−0.510 + 0.884i)13-s + 0.915·16-s + (0.207 + 0.359i)17-s + (0.820 − 1.42i)19-s + (0.562 + 0.975i)20-s + (−0.106 + 0.184i)22-s + (−0.534 − 0.925i)23-s + (−0.171 + 0.297i)25-s + (0.0863 − 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.990 - 0.134i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.990 - 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3469875530\)
\(L(\frac12)\) \(\approx\) \(0.3469875530\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.239T + 2T^{2} \)
5 \( 1 + (1.29 + 2.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.09 + 3.62i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.84 - 3.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.855 - 1.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.57 + 6.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.56 + 4.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.06 + 1.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.53T + 31T^{2} \)
37 \( 1 + (0.830 - 1.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.10 - 8.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.830 - 1.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.33T + 47T^{2} \)
53 \( 1 + (-5.32 - 9.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.06T + 59T^{2} \)
61 \( 1 + 7.98T + 61T^{2} \)
67 \( 1 - 8.26T + 67T^{2} \)
71 \( 1 + 6.23T + 71T^{2} \)
73 \( 1 + (3.57 + 6.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 9.82T + 79T^{2} \)
83 \( 1 + (-3.44 - 5.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.51 - 4.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.53 + 2.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.146523833594017909237234047853, −8.523084952550889571418054657873, −7.893348597166316990065579906592, −6.79915842226836453819795422474, −5.70910417637723843435525979577, −4.72974854025240376903593553548, −4.25807310545790186080432405472, −3.17060439105047676545195904204, −1.32586327695336330610244566686, −0.17404715473448234128596140587, 1.68863663819260655042918334804, 3.34733416117920738847745057888, 3.80937577461872839094286603800, 5.02853451091341806205913196201, 5.76792939465784873279520121787, 7.20765863281452453372565948060, 7.45695875311459317393283768528, 8.363270347457312083070712257321, 9.452569656683456802618927509278, 9.951150594437930005626895395295

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