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Properties

Label 2-1323-63.59-c1-0-24
Degree $2$
Conductor $1323$
Sign $-0.193 - 0.981i$
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  + (2.23 + 1.28i)2-s + (2.32 + 4.02i)4-s + 2.33·5-s + 6.82i·8-s + (5.20 + 3.00i)10-s + 4.36i·11-s + (−1.14 − 0.660i)13-s + (−4.15 + 7.18i)16-s + (2.89 − 5.01i)17-s + (−0.584 + 0.337i)19-s + (5.41 + 9.38i)20-s + (−5.62 + 9.74i)22-s − 5.56i·23-s + 0.437·25-s + (−1.70 − 2.94i)26-s + ⋯
L(s)  = 1  + (1.57 + 0.911i)2-s + (1.16 + 2.01i)4-s + 1.04·5-s + 2.41i·8-s + (1.64 + 0.950i)10-s + 1.31i·11-s + (−0.317 − 0.183i)13-s + (−1.03 + 1.79i)16-s + (0.701 − 1.21i)17-s + (−0.134 + 0.0774i)19-s + (1.21 + 2.09i)20-s + (−1.19 + 2.07i)22-s − 1.16i·23-s + 0.0875·25-s + (−0.333 − 0.578i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.193 - 0.981i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.193 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.820237574\)
\(L(\frac12)\) \(\approx\) \(4.820237574\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 1.28i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 2.33T + 5T^{2} \)
11 \( 1 - 4.36iT - 11T^{2} \)
13 \( 1 + (1.14 + 0.660i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.89 + 5.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.584 - 0.337i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.56iT - 23T^{2} \)
29 \( 1 + (3.86 - 2.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.47 - 2.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.50 + 2.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.29 + 5.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.89 - 6.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.246 - 0.427i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.59 - 2.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.15 + 3.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.77 + 1.02i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.41 - 4.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.17iT - 71T^{2} \)
73 \( 1 + (13.0 + 7.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.30 + 9.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.32 - 9.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.66 - 2.87i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.7 - 7.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.793304157700991240756730328365, −9.034616426362206568592211927734, −7.69153411021603450510135691125, −7.21372981127732217644184651277, −6.40000621517806594721186620005, −5.55090197320982995659690545823, −4.98396958363737498221679324310, −4.15246207341088782110773142727, −2.92604764610600022045489196931, −2.05531044580845720879271659823, 1.35698781004853016023414480626, 2.26967271655607157152552715368, 3.34113747257909383982208939076, 4.02565541798369404210670771816, 5.29695686605757106740724699611, 5.80332104290760758001912430765, 6.29766897011569498984594113129, 7.60111242903835650473002760958, 8.849444313820623146528257531596, 9.797142184178930970997817755960

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