Properties

Label 1-15e2-225.16-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: $\chi_{225} (16, \cdot )$
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

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

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