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Properties

Label 1-405-405.308-r0-0-0
Degree $1$
Conductor $405$
Sign $0.975 - 0.222i$
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  + (−0.448 + 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

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

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.975 - 0.222i$
Analytic conductor: \(1.88081\)
Root analytic conductor: \(1.88081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (308, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ 0.975 - 0.222i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9474204254 - 0.1065095475i\)
\(L(\frac12)\) \(\approx\) \(0.9474204254 - 0.1065095475i\)
\(L(1)\) \(\approx\) \(0.8433512222 + 0.1288819823i\)
\(L(1)\) \(\approx\) \(0.8433512222 + 0.1288819823i\)

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

−24.06773120177925855965226694832, −23.70636499678918380457158126623, −22.22617409279716277204021573722, −21.64517737473447629128743272720, −20.8685710243923560966430897626, −20.13855951804416460721841374349, −19.09239318365542709356541459560, −18.221134358044276640629875162222, −17.81721991681941929023090065242, −16.61082771831139031391824714023, −15.7359618926611168512890290400, −14.46279795840544670900953817909, −13.56597635135190329811645085594, −12.68259687851992705578220745294, −11.588210716095859886694160410455, −11.06472743191941703171699492038, −10.07037783947853552344849887229, −8.94420771742848042858590358184, −8.25151037297674213790791888091, −7.318525991251274091520854058, −5.69710479480947139843166701726, −4.65053603493398497702517627767, −3.56276771826361434037669088380, −2.30731736051914056416934194998, −1.388990715122506086032041625711, 0.71324399558615626618249353012, 2.19326760141808288017991199297, 3.94431070459007432181223807224, 5.10416215271158368033813515818, 5.744910966826561723313816944899, 7.19246916397757052457796825186, 7.813228741891069400402586928674, 8.61749586421245550290674852995, 9.8695371970895444323543310867, 10.591972711753986166883783125827, 11.61494497161037476053539031799, 13.09972568793758804768840351046, 13.840684894862226169435032667689, 14.71501574174564717075040457844, 15.670665832174798693617727505965, 16.28117844487542952182616156842, 17.45335458544235346879365260731, 18.12682421522048210594266137434, 18.59990931055402826685093200696, 20.19345451638454264281086286879, 20.481759446491984139219184588849, 21.880183749648246819533298304071, 22.95519365911987209429601427509, 23.493612008381507459693000130849, 24.47685718355994960484545697432

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