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Properties

Label 1-405-405.178-r1-0-0
Degree $1$
Conductor $405$
Sign $-0.222 - 0.975i$
Analytic cond. $43.5232$
Root an. cond. $43.5232$
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  + (0.448 − 0.893i)2-s + (−0.597 − 0.802i)4-s + (0.918 − 0.396i)7-s + (−0.984 + 0.173i)8-s + (0.973 − 0.230i)11-s + (0.549 − 0.835i)13-s + (0.0581 − 0.998i)14-s + (−0.286 + 0.957i)16-s + (0.342 + 0.939i)17-s + (0.939 + 0.342i)19-s + (0.230 − 0.973i)22-s + (0.918 + 0.396i)23-s + (−0.5 − 0.866i)26-s + (−0.866 − 0.5i)28-s + (0.0581 + 0.998i)29-s + ⋯
L(s)  = 1  + (0.448 − 0.893i)2-s + (−0.597 − 0.802i)4-s + (0.918 − 0.396i)7-s + (−0.984 + 0.173i)8-s + (0.973 − 0.230i)11-s + (0.549 − 0.835i)13-s + (0.0581 − 0.998i)14-s + (−0.286 + 0.957i)16-s + (0.342 + 0.939i)17-s + (0.939 + 0.342i)19-s + (0.230 − 0.973i)22-s + (0.918 + 0.396i)23-s + (−0.5 − 0.866i)26-s + (−0.866 − 0.5i)28-s + (0.0581 + 0.998i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.222 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.222 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.222 - 0.975i$
Analytic conductor: \(43.5232\)
Root analytic conductor: \(43.5232\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (1:\ ),\ -0.222 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.903235012 - 2.385361490i\)
\(L(\frac12)\) \(\approx\) \(1.903235012 - 2.385361490i\)
\(L(1)\) \(\approx\) \(1.321585226 - 0.8918795715i\)
\(L(1)\) \(\approx\) \(1.321585226 - 0.8918795715i\)

Euler product

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

−24.61351622877759719633700384512, −23.54687696674505643735045665588, −22.77229363894549120613295972805, −21.89959701665566896328585208392, −21.11726306277560775980800179844, −20.2881013602629235076987907464, −18.81843323730074198516218671417, −18.13936810125737522703692258561, −17.22360499601273444348262941722, −16.43411919556334346969318885875, −15.52253920867867399832657898327, −14.53867567348668580098993593297, −14.0758742401080749528406204442, −13.01609956761127254190337829597, −11.78551492553335542461032517408, −11.38527568531060446039342826802, −9.48937360873938660619203986090, −8.91084698264302721816338296328, −7.78392067303149900823715512610, −6.94054227861887027844088625362, −5.91229253791907638464248670565, −4.88231637820617773603907516005, −4.08846384034406110017309535214, −2.74405259841490453625342685906, −1.13368954687407811887427042868, 0.92799652665219963268219076857, 1.65296607103442462637869072932, 3.24527373132288084983876647234, 3.983325556253383820865338334, 5.17736821340652746368039436867, 6.000849138854382651638196896980, 7.47217906890890666010048912891, 8.61379441089391490983244351600, 9.5242239677856388592457947779, 10.79330066317575740457537683786, 11.13462563816737620076494769782, 12.29625809296963884989398512155, 13.067422328600059780644860723397, 14.26866098396181255854327654986, 14.55583568188087203568682104233, 15.796167219953215000072594270650, 17.142722723549748630617479868542, 17.86251764735805503095072669894, 18.79139260913617078928769949422, 19.78011073723569140674738550981, 20.423557716134823258524055673507, 21.241635975870125206654482174502, 22.05105114436908620429773730793, 22.9157664547310643337096951101, 23.7256997368704793127271644516

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