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Properties

Label 1-15e2-225.83-r0-0-0
Degree $1$
Conductor $225$
Sign $-0.747 - 0.663i$
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  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (−0.866 + 0.5i)7-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)11-s + (0.406 + 0.913i)13-s + (−0.104 − 0.994i)14-s + (−0.104 + 0.994i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.743 − 0.669i)22-s + (−0.994 + 0.104i)23-s − 26-s + (0.951 + 0.309i)28-s + (−0.978 + 0.207i)29-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (−0.866 + 0.5i)7-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)11-s + (0.406 + 0.913i)13-s + (−0.104 − 0.994i)14-s + (−0.104 + 0.994i)16-s + (−0.951 + 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.743 − 0.669i)22-s + (−0.994 + 0.104i)23-s − 26-s + (0.951 + 0.309i)28-s + (−0.978 + 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.747 - 0.663i$
Analytic conductor: \(1.04489\)
Root analytic conductor: \(1.04489\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (0:\ ),\ -0.747 - 0.663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04545846637 + 0.1196702573i\)
\(L(\frac12)\) \(\approx\) \(-0.04545846637 + 0.1196702573i\)
\(L(1)\) \(\approx\) \(0.4650917649 + 0.2415191704i\)
\(L(1)\) \(\approx\) \(0.4650917649 + 0.2415191704i\)

Euler product

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

−26.023537586117174906598455547919, −25.33363106402293713563516677989, −23.78069957185377777029252452201, −22.74311580413150796688710406028, −22.251223451016781500541712540701, −20.834240420653142626211635616494, −20.32268829872878633566237784086, −19.39068851207909547114590111516, −18.36963757965421185652293740603, −17.65137907930197706216761848216, −16.48018949762316077967705112352, −15.61032162978301838488539767819, −14.04344093373126596748396183634, −12.99712149569868498374891684682, −12.53172825962546111434207525735, −11.03621260680459299107777852990, −10.32612108429932518704780391563, −9.45043800619025504334040031719, −8.21127992130455407265382431094, −7.25692469837720714082996225228, −5.69125661375562383042834080585, −4.21038373515913056916489318245, −3.21099393627511340716089449608, −1.96223565757650370319829223733, −0.10045804857211099947829656010, 2.10729361620338727888467087695, 3.83457627679177024639800744087, 5.21328055546581931349240877537, 6.24013135168546139892750930796, 7.081618809232481757903052426949, 8.44985472283198579858414541670, 9.15146832962960955077185337201, 10.24038356159064302880838159390, 11.372664917165713347125710504375, 12.99737248516644137706754378063, 13.5781736591174601224156968400, 14.907389962412864612090873263495, 15.80811587228407330093335932942, 16.364558230321222639959762342509, 17.54844183171209670163146250485, 18.54743244177279376851492221687, 19.14605116247993081680059555818, 20.25264597607666567649246056634, 21.77766197641403467465694008977, 22.361369595215950170020998092053, 23.7902697534473120470086064572, 24.00268303158189225554982761440, 25.38768079853625510592859260576, 26.08063131631051701335693533259, 26.606324105338803428901303869969

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