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Properties

Label 1-1323-1323.1067-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.878 - 0.477i$
Analytic cond. $6.14398$
Root an. cond. $6.14398$
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  + (−0.270 − 0.962i)2-s + (−0.853 + 0.521i)4-s + (0.542 − 0.840i)5-s + (0.733 + 0.680i)8-s + (−0.955 − 0.294i)10-s + (−0.698 + 0.715i)11-s + (−0.995 + 0.0995i)13-s + (0.456 − 0.889i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (−0.0249 + 0.999i)20-s + (0.878 + 0.478i)22-s + (−0.878 − 0.478i)23-s + (−0.411 − 0.911i)25-s + (0.365 + 0.930i)26-s + ⋯
L(s)  = 1  + (−0.270 − 0.962i)2-s + (−0.853 + 0.521i)4-s + (0.542 − 0.840i)5-s + (0.733 + 0.680i)8-s + (−0.955 − 0.294i)10-s + (−0.698 + 0.715i)11-s + (−0.995 + 0.0995i)13-s + (0.456 − 0.889i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (−0.0249 + 0.999i)20-s + (0.878 + 0.478i)22-s + (−0.878 − 0.478i)23-s + (−0.411 − 0.911i)25-s + (0.365 + 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.878 - 0.477i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.878 - 0.477i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9820005789 - 0.2494274043i\)
\(L(\frac12)\) \(\approx\) \(0.9820005789 - 0.2494274043i\)
\(L(1)\) \(\approx\) \(0.7732812694 - 0.3339315775i\)
\(L(1)\) \(\approx\) \(0.7732812694 - 0.3339315775i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.270 - 0.962i)T \)
5 \( 1 + (0.542 - 0.840i)T \)
11 \( 1 + (-0.698 + 0.715i)T \)
13 \( 1 + (-0.995 + 0.0995i)T \)
17 \( 1 + (-0.988 + 0.149i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.878 - 0.478i)T \)
29 \( 1 + (0.853 + 0.521i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (0.542 - 0.840i)T \)
43 \( 1 + (-0.998 + 0.0498i)T \)
47 \( 1 + (0.698 - 0.715i)T \)
53 \( 1 + (0.988 + 0.149i)T \)
59 \( 1 + (-0.797 - 0.603i)T \)
61 \( 1 + (0.853 + 0.521i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.988 + 0.149i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.995 + 0.0995i)T \)
89 \( 1 + (0.0747 + 0.997i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−21.37219382498115857812947503230, −19.88776177564790229698965440535, −19.39533499075563744108537128258, −18.407997069964124871688155560569, −17.90769027832855366291661038599, −17.33229998441595179367586736477, −16.40871704227148746717784364428, −15.50613249810497325583880679406, −15.13740854773510319282824191269, −13.92917708613241383751650373837, −13.7790755572220443270625778628, −12.82955517944367324578040361385, −11.52672799431683934459276893690, −10.71423389937277426252762482720, −9.88191937603530887617367646724, −9.33992320997133885119532561821, −8.22422339798033324736442916600, −7.53735510395858518019894566696, −6.7059900291216441881570682789, −6.0214651157712048400332183306, −5.190508622048408762322473969536, −4.3187434794592349754257448328, −2.99225507497399327010551995002, −2.17094554258914603282061190866, −0.545771928146028733047650186427, 0.902423297048817303629353048861, 2.06903872644400607436923854345, 2.52930376665241989934636803684, 3.93555071952187050625224635038, 4.75857717506756812418925478561, 5.30792114732017952548464344589, 6.61654707270902268767797086456, 7.801632625171175370089136853487, 8.44092602221419428617834234703, 9.32549722256758041786143335566, 10.085828936371606773234161365711, 10.465152021635380897481684841717, 11.89279970044735963577206455599, 12.221748745844601654448692388619, 13.030873057600822264999173765779, 13.71873799853531107329151233093, 14.50967660721871564649439625669, 15.67097191551803239242013228520, 16.53996592427504270869405477013, 17.30411875296863006351232414710, 17.88962253543359362753979908536, 18.54182701431129242839908267243, 19.62985204667039394790799589076, 20.19234219494054571665706536384, 20.68508043740261621471000663400

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