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Properties

Label 1-15e2-225.211-r0-0-0
Degree $1$
Conductor $225$
Sign $-0.658 + 0.752i$
Analytic cond. $1.04489$
Root an. cond. $1.04489$
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  + (0.669 + 0.743i)2-s + (−0.104 + 0.994i)4-s + (−0.5 + 0.866i)7-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)11-s + (0.669 − 0.743i)13-s + (−0.978 + 0.207i)14-s + (−0.978 − 0.207i)16-s + (−0.809 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.104 + 0.994i)22-s + (−0.978 + 0.207i)23-s + 26-s + (−0.809 − 0.587i)28-s + (0.913 − 0.406i)29-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (−0.104 + 0.994i)4-s + (−0.5 + 0.866i)7-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)11-s + (0.669 − 0.743i)13-s + (−0.978 + 0.207i)14-s + (−0.978 − 0.207i)16-s + (−0.809 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.104 + 0.994i)22-s + (−0.978 + 0.207i)23-s + 26-s + (−0.809 − 0.587i)28-s + (0.913 − 0.406i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(1.04489\)
Root analytic conductor: \(1.04489\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (0:\ ),\ -0.658 + 0.752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6006756214 + 1.324182057i\)
\(L(\frac12)\) \(\approx\) \(0.6006756214 + 1.324182057i\)
\(L(1)\) \(\approx\) \(1.028879006 + 0.8369493514i\)
\(L(1)\) \(\approx\) \(1.028879006 + 0.8369493514i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.669 + 0.743i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.669 - 0.743i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.978 + 0.207i)T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (0.913 + 0.406i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−26.27593868478430169536538660800, −24.95718910628205374902619815735, −23.92379022012345822801732007317, −23.285126021742330262402563301299, −22.25990608921744217365091544860, −21.538272289706539029902323523980, −20.460120666628645539219338548263, −19.66262574240741326268643112772, −18.94627071728709042696826164456, −17.73178666213430673033859971730, −16.45373050937762896675826073785, −15.58904663599516515312559694414, −14.11185372644174663080306344666, −13.74011345956584242292873041508, −12.66789653830284849882623543048, −11.476053301529957589863843836298, −10.79672302862459015216760992229, −9.65001786269676346804547970110, −8.65170274515956896147398090475, −6.79789244978775369785301929268, −6.13260715094846917211461754516, −4.48148340366476285503750741240, −3.79259785656296315541265177943, −2.4487599474775518744667328909, −0.88647056984296021228264334178, 2.217132283833823592086517560637, 3.56142429829231002347948213480, 4.63500721336503486194404051262, 6.0233799976182667473408296774, 6.527136612293987927889796361739, 8.04759680653657946118019804497, 8.82225199645707363668397127748, 10.134115741469915706802945799977, 11.70169637999337385813908129125, 12.50739906357817815029609422152, 13.33071966393091436299381518232, 14.54036962770531742956945165954, 15.3876660443267222599253529522, 16.02774895717290494683060602660, 17.311258567745428831921498053297, 17.98014754336955062234887735265, 19.277633784718601370679520348389, 20.409382843249657766232938237521, 21.47815431415996956363305857087, 22.31437258003690248868874361228, 22.9818513741607851905972660150, 23.99122928783037866173461427972, 25.19937383495800055282085499038, 25.37545187608466524282038202678, 26.51066313778163550861495940955

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