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Properties

Label 2-1323-9.7-c1-0-11
Degree $2$
Conductor $1323$
Sign $-0.0910 - 0.995i$
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 + (−0.0527 − 0.0913i)5-s − 3.07·8-s + 0.116·10-s + (1.66 − 2.89i)11-s + (1.23 + 2.14i)13-s + (0.909 − 1.57i)16-s + 1.61·17-s + 7.68·19-s + (0.0413 − 0.0715i)20-s + (1.84 + 3.18i)22-s + (−0.948 − 1.64i)23-s + (2.49 − 4.32i)25-s − 2.73·26-s + ⋯
L(s)  = 1  + (−0.389 + 0.675i)2-s + (0.195 + 0.339i)4-s + (−0.0235 − 0.0408i)5-s − 1.08·8-s + 0.0367·10-s + (0.503 − 0.871i)11-s + (0.343 + 0.595i)13-s + (0.227 − 0.393i)16-s + 0.391·17-s + 1.76·19-s + (0.00924 − 0.0160i)20-s + (0.392 + 0.679i)22-s + (−0.197 − 0.342i)23-s + (0.498 − 0.864i)25-s − 0.536·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0910 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0910 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.0910 - 0.995i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.0910 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.457502991\)
\(L(\frac12)\) \(\approx\) \(1.457502991\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0527 + 0.0913i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.66 + 2.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.23 - 2.14i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
19 \( 1 - 7.68T + 19T^{2} \)
23 \( 1 + (0.948 + 1.64i)T + (-11.5 + 19.9i)T^{2} \)
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 \( 1 + 1.98T + 37T^{2} \)
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 \( 1 - 9.97T + 53T^{2} \)
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 \( 1 - 4.72T + 73T^{2} \)
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 \( 1 - 6.02T + 89T^{2} \)
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

−9.548145297975253472117787461315, −8.844896782157880567982633893533, −8.281319700595015090188878041978, −7.36886109781761939178673304156, −6.67840120206321138826044045622, −5.94180981683131358670356412682, −4.94979954524610475178397425857, −3.59016300936608875503683322102, −2.93772196526725814365017542113, −1.18742921018702911078882951118, 0.824309466962494190122408935265, 1.94604203673206829942396245372, 3.04517276242813721529732515142, 4.02436039687348181535276619568, 5.39874594594583670848925149849, 5.92961046830435343143951512605, 7.12298693902829631281694016092, 7.73964168043679762934222262022, 8.940648305663515292973045628025, 9.628663712929349242618326823111

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