↑

Properties

Label 1-1083-1083.65-r0-0-0
Degree $1$
Conductor $1083$
Sign $-0.161 + 0.986i$
Analytic cond. $5.02943$
Root an. cond. $5.02943$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.635 + 0.771i)2-s + (−0.191 + 0.981i)4-s + (0.926 + 0.376i)5-s + (0.945 + 0.324i)7-s + (−0.879 + 0.475i)8-s + (0.298 + 0.954i)10-s + (0.401 − 0.915i)11-s + (−0.451 − 0.892i)13-s + (0.350 + 0.936i)14-s + (−0.926 − 0.376i)16-s + (0.191 + 0.981i)17-s + (−0.546 + 0.837i)20-s + (0.962 − 0.272i)22-s + (0.998 − 0.0550i)23-s + (0.716 + 0.697i)25-s + (0.401 − 0.915i)26-s + ⋯
L(s)  = 1  + (0.635 + 0.771i)2-s + (−0.191 + 0.981i)4-s + (0.926 + 0.376i)5-s + (0.945 + 0.324i)7-s + (−0.879 + 0.475i)8-s + (0.298 + 0.954i)10-s + (0.401 − 0.915i)11-s + (−0.451 − 0.892i)13-s + (0.350 + 0.936i)14-s + (−0.926 − 0.376i)16-s + (0.191 + 0.981i)17-s + (−0.546 + 0.837i)20-s + (0.962 − 0.272i)22-s + (0.998 − 0.0550i)23-s + (0.716 + 0.697i)25-s + (0.401 − 0.915i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $-0.161 + 0.986i$
Analytic conductor: \(5.02943\)
Root analytic conductor: \(5.02943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1083} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1083,\ (0:\ ),\ -0.161 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.802899900 + 2.122430278i\)
\(L(\frac12)\) \(\approx\) \(1.802899900 + 2.122430278i\)
\(L(1)\) \(\approx\) \(1.537250103 + 0.9919900515i\)
\(L(1)\) \(\approx\) \(1.537250103 + 0.9919900515i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.635 + 0.771i)T \)
5 \( 1 + (0.926 + 0.376i)T \)
7 \( 1 + (0.945 + 0.324i)T \)
11 \( 1 + (0.401 - 0.915i)T \)
13 \( 1 + (-0.451 - 0.892i)T \)
17 \( 1 + (0.191 + 0.981i)T \)
23 \( 1 + (0.998 - 0.0550i)T \)
29 \( 1 + (-0.754 + 0.656i)T \)
31 \( 1 + (0.986 + 0.164i)T \)
37 \( 1 + (0.401 - 0.915i)T \)
41 \( 1 + (-0.821 + 0.569i)T \)
43 \( 1 + (-0.298 + 0.954i)T \)
47 \( 1 + (0.592 + 0.805i)T \)
53 \( 1 + (-0.592 - 0.805i)T \)
59 \( 1 + (-0.821 + 0.569i)T \)
61 \( 1 + (0.851 - 0.523i)T \)
67 \( 1 + (-0.0275 + 0.999i)T \)
71 \( 1 + (0.851 + 0.523i)T \)
73 \( 1 + (-0.191 - 0.981i)T \)
79 \( 1 + (0.298 - 0.954i)T \)
83 \( 1 + (-0.789 - 0.614i)T \)
89 \( 1 + (-0.191 + 0.981i)T \)
97 \( 1 + (-0.0275 - 0.999i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.1446993769232157318637436347, −20.56217431990305530693258435245, −20.09795524656308705182858495880, −18.88566130471444966036077566585, −18.319665994716877043005500772778, −17.24077044887676570656750041178, −16.92935055484222294762649388771, −15.427671586844865050642290061014, −14.75583195183259624271283968576, −13.87009481265851571639300812674, −13.600019312923211264089383018915, −12.43423679403935781741524199005, −11.82157915928817710975112091499, −11.058476526921525088814936329016, −10.00173403080483824474875975496, −9.52344094714817463925505728435, −8.66734605995179509542406587079, −7.25669724990544039758403238792, −6.49453308990635925952715479006, −5.24663448110151399379610901606, −4.82731850882095327794222776393, −4.00717550377466175483325005964, −2.586972385340124157003299329023, −1.880506104435037369954163092381, −1.05317834823193025765010703969, 1.38352632754729028383862302421, 2.63211673998199522333042890171, 3.38527568606933095280970367857, 4.63800820716003116745787526356, 5.470495590595553875014853434195, 6.003545852611816605901604341500, 6.92297252610980215484733039964, 7.94290876689130884309194759749, 8.59210874165781185348258092305, 9.49532982198090918401848367409, 10.71050878406604195790895196435, 11.35512962982680379909191655532, 12.47320724875962976751220778110, 13.14501426769090816396543062682, 13.972591698513731664819778462686, 14.75508859103746171597823542719, 15.0055602699502398729296450617, 16.22228960014601715798575484607, 17.13330510428791457799449593876, 17.50337714568245202804090614782, 18.305824859475295903810045533774, 19.18373249446770944763907356799, 20.45611901081977475038507661410, 21.22004397924536365419615437158, 21.74531090901531272241522449570

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