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Properties

Label 1-405-405.4-r0-0-0
Degree $1$
Conductor $405$
Sign $0.657 - 0.753i$
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  + (−0.973 − 0.230i)2-s + (0.893 + 0.448i)4-s + (0.835 + 0.549i)7-s + (−0.766 − 0.642i)8-s + (−0.993 + 0.116i)11-s + (0.286 − 0.957i)13-s + (−0.686 − 0.727i)14-s + (0.597 + 0.802i)16-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (0.993 + 0.116i)22-s + (0.835 − 0.549i)23-s + (−0.5 + 0.866i)26-s + (0.5 + 0.866i)28-s + (−0.686 + 0.727i)29-s + ⋯
L(s)  = 1  + (−0.973 − 0.230i)2-s + (0.893 + 0.448i)4-s + (0.835 + 0.549i)7-s + (−0.766 − 0.642i)8-s + (−0.993 + 0.116i)11-s + (0.286 − 0.957i)13-s + (−0.686 − 0.727i)14-s + (0.597 + 0.802i)16-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (0.993 + 0.116i)22-s + (0.835 − 0.549i)23-s + (−0.5 + 0.866i)26-s + (0.5 + 0.866i)28-s + (−0.686 + 0.727i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(1.88081\)
Root analytic conductor: \(1.88081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ 0.657 - 0.753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7765933600 - 0.3530049901i\)
\(L(\frac12)\) \(\approx\) \(0.7765933600 - 0.3530049901i\)
\(L(1)\) \(\approx\) \(0.7443511798 - 0.1334375632i\)
\(L(1)\) \(\approx\) \(0.7443511798 - 0.1334375632i\)

Euler product

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

−24.46098814608555555105830654418, −23.754480594155467846198656500515, −23.13903145372427696431563561980, −21.34393861912511616119044798120, −21.035259134725542190559864248975, −20.014488263964484306902150045805, −19.05214460239081861615094354162, −18.35173626135096954445449508491, −17.45063874252430290235614321446, −16.73123240281920812992698282233, −15.85747558785687926546235108698, −14.87555273307623368915456718442, −14.07592004570116206852737446131, −12.86509193079182899352294749286, −11.55786717605993081083129396288, −10.88924289722871812523623355509, −10.08114194547226321877927035984, −8.95390235265152536999684160701, −8.036995360510211988733855880329, −7.374610588446791442863410624134, −6.20132369323868598575072233849, −5.16140721165373313364567384160, −3.79315593761323611758838431271, −2.22804974643035988852983469977, −1.24313879926200403376455247122, 0.77657831677574592789236073897, 2.2907580672094470814000593242, 3.012790142966880962445672792358, 4.78168307721631935480025919062, 5.77392608892199790003436983188, 7.17215858312975341718571118950, 7.92354823547198988850628251532, 8.79856538425668597271197358556, 9.69808900277354033622284227871, 10.89853112413593522305302835827, 11.30521378479978664995499781620, 12.51951425560216346300193844771, 13.34808054773992751993185809167, 14.909730093543673770422493213965, 15.47242324186374402913860360178, 16.41736307823473905121099508359, 17.50169499412397291671855910407, 18.243223256405288710655121092998, 18.638877514476937230377644533897, 20.02050591893321624564483852582, 20.58056854428486312331093084427, 21.35196554332518334153030191182, 22.31439535264308299930728606894, 23.52578585795760899363551579795, 24.541423181291206149155531829099

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