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Properties

Label 1-1323-1323.401-r1-0-0
Degree $1$
Conductor $1323$
Sign $0.576 - 0.817i$
Analytic cond. $142.176$
Root an. cond. $142.176$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.583 − 0.811i)2-s + (−0.318 − 0.947i)4-s + (−0.921 + 0.388i)5-s + (−0.955 − 0.294i)8-s + (−0.222 + 0.974i)10-s + (0.583 − 0.811i)11-s + (−0.969 + 0.246i)13-s + (−0.797 + 0.603i)16-s + (−0.623 + 0.781i)17-s + 19-s + (0.661 + 0.749i)20-s + (−0.318 − 0.947i)22-s + (−0.980 − 0.198i)23-s + (0.698 − 0.715i)25-s + (−0.365 + 0.930i)26-s + ⋯
L(s)  = 1  + (0.583 − 0.811i)2-s + (−0.318 − 0.947i)4-s + (−0.921 + 0.388i)5-s + (−0.955 − 0.294i)8-s + (−0.222 + 0.974i)10-s + (0.583 − 0.811i)11-s + (−0.969 + 0.246i)13-s + (−0.797 + 0.603i)16-s + (−0.623 + 0.781i)17-s + 19-s + (0.661 + 0.749i)20-s + (−0.318 − 0.947i)22-s + (−0.980 − 0.198i)23-s + (0.698 − 0.715i)25-s + (−0.365 + 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.576 - 0.817i$
Analytic conductor: \(142.176\)
Root analytic conductor: \(142.176\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (1:\ ),\ 0.576 - 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.274959700 - 0.6612989011i\)
\(L(\frac12)\) \(\approx\) \(1.274959700 - 0.6612989011i\)
\(L(1)\) \(\approx\) \(0.9076437346 - 0.4368785908i\)
\(L(1)\) \(\approx\) \(0.9076437346 - 0.4368785908i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.583 - 0.811i)T \)
5 \( 1 + (-0.921 + 0.388i)T \)
11 \( 1 + (0.583 - 0.811i)T \)
13 \( 1 + (-0.969 + 0.246i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.980 - 0.198i)T \)
29 \( 1 + (-0.980 + 0.198i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (0.797 + 0.603i)T \)
43 \( 1 + (-0.124 - 0.992i)T \)
47 \( 1 + (-0.995 - 0.0995i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
59 \( 1 + (-0.456 - 0.889i)T \)
61 \( 1 + (-0.318 + 0.947i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.365 + 0.930i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.969 + 0.246i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.86984931180894119213882327534, −20.01053726900620422878562411433, −19.69241261755186860873934690538, −18.24646985283553872352318838203, −17.80249688706689481039712113843, −16.77669242078038911528186451945, −16.241038011993514157482841201529, −15.47946839752602276583102066952, −14.826014785022920697561826465481, −14.15282226876686901632473093753, −13.171326805638375409894601293707, −12.34590348783202229851937264685, −11.92378969318292443383075348709, −11.059826401288851939547461737539, −9.49775842789213957014533612848, −9.16740069179869321941405470740, −7.81066558973132144003297962508, −7.52533635291914504474016335198, −6.721251128643681929484887301077, −5.53707865938912995204378666188, −4.80272609260949672571263449345, −4.09969156324674476155743828355, −3.266558583546354850702463336993, −2.078534812692738395169954672102, −0.42432322697998288480188607169, 0.51646111840647240986218030852, 1.74431182990939955812730722107, 2.767565725510652221989325745664, 3.6765751832277509256698685954, 4.1977301138373393816945037737, 5.25823417796800968404068099555, 6.18788853935051474812604344269, 7.06088114349042493805051271969, 8.07390301855968250767083116523, 9.027756061489332646175643179267, 9.864925247813966485716109896869, 10.7634785470486586592679917457, 11.50488724179500809989334657566, 11.92504323912640638844330997890, 12.8258124627152752999632316867, 13.67953947480035717911226871475, 14.57751001875141206209867988971, 14.930872117062236392332257846340, 15.94659351934485830827567567332, 16.70281692617326204785934536274, 17.86757904332197556245715157355, 18.65708161820356297377085565085, 19.273541961279814851563306726554, 19.96801583206548813689256864602, 20.368858445685269267492515789249

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