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Properties

Label 1-405-405.184-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  + (0.0581 − 0.998i)2-s + (−0.993 − 0.116i)4-s + (−0.597 − 0.802i)7-s + (−0.173 + 0.984i)8-s + (−0.686 + 0.727i)11-s + (−0.893 − 0.448i)13-s + (−0.835 + 0.549i)14-s + (0.973 + 0.230i)16-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)19-s + (0.686 + 0.727i)22-s + (−0.597 + 0.802i)23-s + (−0.5 + 0.866i)26-s + (0.5 + 0.866i)28-s + (−0.835 − 0.549i)29-s + ⋯
L(s)  = 1  + (0.0581 − 0.998i)2-s + (−0.993 − 0.116i)4-s + (−0.597 − 0.802i)7-s + (−0.173 + 0.984i)8-s + (−0.686 + 0.727i)11-s + (−0.893 − 0.448i)13-s + (−0.835 + 0.549i)14-s + (0.973 + 0.230i)16-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)19-s + (0.686 + 0.727i)22-s + (−0.597 + 0.802i)23-s + (−0.5 + 0.866i)26-s + (0.5 + 0.866i)28-s + (−0.835 − 0.549i)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: $\chi_{405} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ 0.323 + 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2094456245 + 0.1497027350i\)
\(L(\frac12)\) \(\approx\) \(0.2094456245 + 0.1497027350i\)
\(L(1)\) \(\approx\) \(0.6062542546 - 0.2691988241i\)
\(L(1)\) \(\approx\) \(0.6062542546 - 0.2691988241i\)

Euler product

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

−24.28711710709864377489192006806, −23.45572346414038360208969390424, −22.49647712070177408747901435687, −21.79961470226283724183706418955, −21.02169062547511045953884609458, −19.45413405948181503138347183361, −18.779500230973089775116546142045, −18.11261034042727856377260256112, −16.792138216516891180387419548748, −16.39215751823023319348458728579, −15.34223861823722882795687854156, −14.66087579962733807858178312592, −13.64466610592404306533334332252, −12.73984954188838039647373966911, −11.95118565924478986636292806621, −10.42758057559595862271463987451, −9.473413494323381641219380132463, −8.63316195517815651602033992116, −7.702155162261179930860261714478, −6.63174709217302153052364855262, −5.74745719595277691477848323093, −4.9192975131858140770891861070, −3.60631594836131233203501082393, −2.40081638894987026088675753803, −0.14513308339967130304758604913, 1.526841126112346341089852454358, 2.75439345535997812641355546171, 3.76006271062136827676948790676, 4.75185199424987944380192574826, 5.842085559296168917441762580135, 7.33780637075381370057038357895, 8.16803759245871430852097233746, 9.63757145096773549472402129426, 10.10609798247156173301887509778, 10.898831692011680093628774975194, 12.275316233873758764876462312438, 12.69550291888912785116801874723, 13.69577502166645624009896119585, 14.58277593058785183622232373614, 15.624372351355795835286825904818, 17.01695615523323327454517896609, 17.4831271630565430034356132590, 18.68976231683643620879759716982, 19.42898431003188647225416332976, 20.16425246688890016985397087066, 20.96170325519037057700322151080, 21.79102073376567533567567244965, 22.91408119212048434050555095439, 23.2001096123222753764575557130, 24.27083743032707892430242354477

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