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Properties

Label 1-405-405.272-r0-0-0
Degree $1$
Conductor $405$
Sign $0.871 - 0.490i$
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  + (0.918 − 0.396i)2-s + (0.686 − 0.727i)4-s + (0.230 + 0.973i)7-s + (0.342 − 0.939i)8-s + (0.835 − 0.549i)11-s + (0.116 + 0.993i)13-s + (0.597 + 0.802i)14-s + (−0.0581 − 0.998i)16-s + (0.642 − 0.766i)17-s + (−0.766 + 0.642i)19-s + (0.549 − 0.835i)22-s + (−0.230 + 0.973i)23-s + (0.5 + 0.866i)26-s + (0.866 + 0.5i)28-s + (0.597 − 0.802i)29-s + ⋯
L(s)  = 1  + (0.918 − 0.396i)2-s + (0.686 − 0.727i)4-s + (0.230 + 0.973i)7-s + (0.342 − 0.939i)8-s + (0.835 − 0.549i)11-s + (0.116 + 0.993i)13-s + (0.597 + 0.802i)14-s + (−0.0581 − 0.998i)16-s + (0.642 − 0.766i)17-s + (−0.766 + 0.642i)19-s + (0.549 − 0.835i)22-s + (−0.230 + 0.973i)23-s + (0.5 + 0.866i)26-s + (0.866 + 0.5i)28-s + (0.597 − 0.802i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.871 - 0.490i$
Analytic conductor: \(1.88081\)
Root analytic conductor: \(1.88081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ 0.871 - 0.490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.391178094 - 0.6271719102i\)
\(L(\frac12)\) \(\approx\) \(2.391178094 - 0.6271719102i\)
\(L(1)\) \(\approx\) \(1.850887569 - 0.3756742122i\)
\(L(1)\) \(\approx\) \(1.850887569 - 0.3756742122i\)

Euler product

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

−24.225071868136908848871951199770, −23.47686796492238705007346126340, −22.82842704562736236399959096773, −21.95793807913636301961512390411, −21.06570225387040138693870009858, −20.11555081638422502590813274841, −19.660822251590510044021369664761, −17.98733284596135230711043420366, −17.17189735236881500594544962823, −16.559149592583427417802954881627, −15.39878960201929308161853090947, −14.60279859913842242349288748496, −13.95174508663029826840101230136, −12.80785717561615118249475864287, −12.29797774918940103919150884450, −10.946485406075554153123846608384, −10.35311729601658938730765372336, −8.74570619248393867791947690160, −7.75063554693187051331065292609, −6.88547319307798360937501907909, −5.991846478801093720021080450162, −4.7287433738515859818006503506, −4.00936536416276268544400828866, −2.90942924066105285240381462745, −1.418836104631623239033877979748, 1.43566892840039279175503983426, 2.474633289450488917365845768381, 3.65140348012168539974166674122, 4.61371862326852181729967174819, 5.803710592077129799147434938843, 6.38476991388791067270863509967, 7.74955920087682480031256791339, 9.07427954073589645822108245936, 9.860225695247060200331538691737, 11.32637805706331850226673471114, 11.70008536017722506795456539549, 12.58383391461574934144460517797, 13.769987470118623777998665556707, 14.37820617978099900227223475987, 15.27416417497146517904912098291, 16.1768025224185504868269232031, 17.0897114581009901475315037638, 18.66987657414675664410026041284, 18.97271400624172138124879565172, 20.065333604083360271668346203, 21.13265536653542774126405343677, 21.613812421148158926514128046348, 22.39518352898555290441692285874, 23.388010411842690875636929199567, 24.12132745030577233474740517697

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