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Properties

Label 1-33e2-1089.166-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.999 + 0.00576i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (−0.786 − 0.618i)2-s + (0.235 + 0.971i)4-s + (0.580 − 0.814i)5-s + (0.0475 + 0.998i)7-s + (0.415 − 0.909i)8-s + (−0.959 + 0.281i)10-s + (0.235 + 0.971i)13-s + (0.580 − 0.814i)14-s + (−0.888 + 0.458i)16-s + (−0.654 + 0.755i)17-s + (−0.654 − 0.755i)19-s + (0.928 + 0.371i)20-s + (0.0475 − 0.998i)23-s + (−0.327 − 0.945i)25-s + (0.415 − 0.909i)26-s + ⋯
L(s)  = 1  + (−0.786 − 0.618i)2-s + (0.235 + 0.971i)4-s + (0.580 − 0.814i)5-s + (0.0475 + 0.998i)7-s + (0.415 − 0.909i)8-s + (−0.959 + 0.281i)10-s + (0.235 + 0.971i)13-s + (0.580 − 0.814i)14-s + (−0.888 + 0.458i)16-s + (−0.654 + 0.755i)17-s + (−0.654 − 0.755i)19-s + (0.928 + 0.371i)20-s + (0.0475 − 0.998i)23-s + (−0.327 − 0.945i)25-s + (0.415 − 0.909i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.00576i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.999 + 0.00576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076967915 + 0.003106890613i\)
\(L(\frac12)\) \(\approx\) \(1.076967915 + 0.003106890613i\)
\(L(1)\) \(\approx\) \(0.8225321345 - 0.1286798463i\)
\(L(1)\) \(\approx\) \(0.8225321345 - 0.1286798463i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.786 - 0.618i)T \)
5 \( 1 + (0.580 - 0.814i)T \)
7 \( 1 + (0.0475 + 0.998i)T \)
13 \( 1 + (0.235 + 0.971i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (0.0475 - 0.998i)T \)
29 \( 1 + (0.981 - 0.189i)T \)
31 \( 1 + (0.723 + 0.690i)T \)
37 \( 1 + (-0.959 + 0.281i)T \)
41 \( 1 + (0.928 - 0.371i)T \)
43 \( 1 + (0.580 + 0.814i)T \)
47 \( 1 + (0.928 + 0.371i)T \)
53 \( 1 + (0.841 - 0.540i)T \)
59 \( 1 + (-0.786 + 0.618i)T \)
61 \( 1 + (0.928 + 0.371i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (-0.995 - 0.0950i)T \)
83 \( 1 + (0.0475 + 0.998i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.580 + 0.814i)T \)
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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

−21.29566646539071744345128998800, −20.48527583120273839839992781873, −19.76714910203410373679836569464, −18.94455586177698064120976380939, −18.19755707125625854749441924855, −17.38475806164627292855715003331, −17.168482438789631170744959388845, −15.891333347727998771180877390172, −15.40052114436606977138575564021, −14.33801408611829831092350554659, −13.88592302985536498526066721449, −13.056212020680005168750730803357, −11.5999410684996349628674765583, −10.70954942423179731589434901186, −10.303777514424293077065426504719, −9.539524193756670163965357424120, −8.500572396286652344755113076527, −7.571562491402242252223054595753, −6.996821133752405620953930526947, −6.13838817763304107199159542591, −5.37174485080845089295949048892, −4.157084598207433286377022999013, −2.9278385325703893272338152873, −1.87705333835215286294373958162, −0.70186044168955633655531102195, 1.013296835557002171238041671046, 2.09362525301140974597530020277, 2.60569061058642116014567503554, 4.12094393248528135696484785388, 4.82137848492229865929441026328, 6.152412727514926059988673535490, 6.77316397597432465454763561409, 8.26289186904397425162444106054, 8.74916473831210320639771736917, 9.23172126670565491878514130619, 10.24377274146597312692738758300, 11.0448675352069536970701008962, 12.02929176769545162700711827081, 12.52830082569488874956285917365, 13.3105802192542304912117316562, 14.2542076961980824935444615571, 15.53020581649576925647225710713, 16.08605124567838470591173265575, 17.02510510827626609143616622741, 17.59524577197638162340804080342, 18.31020034245438181739790354693, 19.24994328410237000539560864858, 19.67581106092979800867627799425, 20.81681544769373383081541281395, 21.30810650748122197282326893136

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