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Properties

Label 1-405-405.259-r0-0-0
Degree $1$
Conductor $405$
Sign $0.323 - 0.946i$
Analytic cond. $1.88081$
Root an. cond. $1.88081$
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  + (−0.893 − 0.448i)2-s + (0.597 + 0.802i)4-s + (−0.396 − 0.918i)7-s + (−0.173 − 0.984i)8-s + (0.973 − 0.230i)11-s + (0.835 + 0.549i)13-s + (−0.0581 + 0.998i)14-s + (−0.286 + 0.957i)16-s + (0.939 − 0.342i)17-s + (−0.939 − 0.342i)19-s + (−0.973 − 0.230i)22-s + (−0.396 + 0.918i)23-s + (−0.5 − 0.866i)26-s + (0.5 − 0.866i)28-s + (−0.0581 − 0.998i)29-s + ⋯
L(s)  = 1  + (−0.893 − 0.448i)2-s + (0.597 + 0.802i)4-s + (−0.396 − 0.918i)7-s + (−0.173 − 0.984i)8-s + (0.973 − 0.230i)11-s + (0.835 + 0.549i)13-s + (−0.0581 + 0.998i)14-s + (−0.286 + 0.957i)16-s + (0.939 − 0.342i)17-s + (−0.939 − 0.342i)19-s + (−0.973 − 0.230i)22-s + (−0.396 + 0.918i)23-s + (−0.5 − 0.866i)26-s + (0.5 − 0.866i)28-s + (−0.0581 − 0.998i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.323 - 0.946i$
Analytic conductor: \(1.88081\)
Root analytic conductor: \(1.88081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ 0.323 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6997697477 - 0.5001653549i\)
\(L(\frac12)\) \(\approx\) \(0.6997697477 - 0.5001653549i\)
\(L(1)\) \(\approx\) \(0.7201988423 - 0.2418324114i\)
\(L(1)\) \(\approx\) \(0.7201988423 - 0.2418324114i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.893 - 0.448i)T \)
7 \( 1 + (-0.396 - 0.918i)T \)
11 \( 1 + (0.973 - 0.230i)T \)
13 \( 1 + (0.835 + 0.549i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.396 + 0.918i)T \)
29 \( 1 + (-0.0581 - 0.998i)T \)
31 \( 1 + (-0.993 + 0.116i)T \)
37 \( 1 + (-0.766 - 0.642i)T \)
41 \( 1 + (0.893 - 0.448i)T \)
43 \( 1 + (0.686 + 0.727i)T \)
47 \( 1 + (0.993 + 0.116i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.973 + 0.230i)T \)
61 \( 1 + (0.597 - 0.802i)T \)
67 \( 1 + (0.0581 - 0.998i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (-0.173 - 0.984i)T \)
79 \( 1 + (0.893 + 0.448i)T \)
83 \( 1 + (-0.893 - 0.448i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.286 - 0.957i)T \)
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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

−24.80231275756911994811535754496, −23.8120791761930824007924405836, −22.90363524475252075655785598616, −21.97624726793861172528744978500, −20.83467861264218033490466169346, −20.007462411444700103623738129194, −19.00382730942265984763104420008, −18.52131146164297630991492857009, −17.51990745425450015574412982096, −16.625524196656352278563385548659, −15.89579150699302133901752495418, −14.905127820605380485900584088183, −14.330466819171189335217191886762, −12.75950112676100020908266638256, −11.97105856690630061733579098714, −10.81798391912823081591299336604, −9.99333560520117706475786836578, −8.89178822447701299323101878534, −8.4322022720158810327323148582, −7.132244859806062460469808259963, −6.15445928542698459276757617434, −5.50959726878506995038413366020, −3.83398726671545438843143372323, −2.43991323649716231589745395179, −1.230528548198309437776779799700, 0.79689618781054678684180444548, 1.949535468308233311795953477513, 3.509606630429823675775770117110, 4.061065105506557540531774144839, 6.01368230485006344160708339968, 6.93328434390357706870332922421, 7.80356275920378632826108203529, 8.97018753383499267460254130890, 9.64025260518593015842313833039, 10.70322076913237629609953789120, 11.40407810372317149281668974286, 12.3980149249088571133727187479, 13.43387901533886258992163793960, 14.33164443244293245773060870468, 15.736347819266173618986594202535, 16.52923990187494608243555618303, 17.1595481899365533840484468560, 18.05363760175903325984452210989, 19.311673337452260316931058711017, 19.4453907757055373322463135399, 20.68050058390004164697123196378, 21.26056735378223688450870593005, 22.33911958096796470924106976895, 23.29642635601517035068516412790, 24.24534685114011904478248624164

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