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Properties

Label 1-15e2-225.166-r0-0-0
Degree $1$
Conductor $225$
Sign $-0.988 - 0.152i$
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  + (−0.978 − 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.5 + 0.866i)7-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)11-s + (−0.978 + 0.207i)13-s + (0.669 − 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.913 + 0.406i)22-s + (0.669 − 0.743i)23-s + 26-s + (−0.809 + 0.587i)28-s + (−0.104 − 0.994i)29-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.5 + 0.866i)7-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)11-s + (−0.978 + 0.207i)13-s + (0.669 − 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.913 + 0.406i)22-s + (0.669 − 0.743i)23-s + 26-s + (−0.809 + 0.587i)28-s + (−0.104 − 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.988 - 0.152i$
Analytic conductor: \(1.04489\)
Root analytic conductor: \(1.04489\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (0:\ ),\ -0.988 - 0.152i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003682530902 - 0.04785876021i\)
\(L(\frac12)\) \(\approx\) \(0.003682530902 - 0.04785876021i\)
\(L(1)\) \(\approx\) \(0.4497149501 + 0.02187325199i\)
\(L(1)\) \(\approx\) \(0.4497149501 + 0.02187325199i\)

Euler product

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

−26.89049178842444893682904387013, −25.97760949093656032815678970316, −25.37509403993598517900602349038, −24.05013247709283485850273414685, −23.53711477432799845847007317790, −22.23819070187498998279427149195, −20.9720794354868003160319690280, −20.10945625274005024368074297439, −19.3504418711323371603857855262, −18.41272359370399536489596098100, −17.262187339785023525273281801825, −16.78298789564415807188667058746, −15.56576651606077685324657341815, −14.83927028008573553524133688506, −13.39392781089713327616566282499, −12.39857738413981580195831565367, −10.94966189852886366835935779966, −10.295085194321804315252190638061, −9.36705054235741595840792450908, −8.0765454911533506714662598427, −7.264793633698721231594014673, −6.25793468460185609611666722584, −4.855711616044076558594556186260, −3.17483665138496303389873350177, −1.820523768668622563873579880865, 0.043433541409009698715785672319, 2.226329997467569611636559753142, 2.92603755892893345832239366880, 4.84811113219482193295828303600, 6.263826886562932073455655049253, 7.2473380000373799819856932359, 8.477564859011710338079699623070, 9.26183226889311903055706251199, 10.28529633863906299500889417536, 11.29919240465512727015516028765, 12.34252390851218241306892508456, 13.19245786830347050845360388951, 14.92676401588575666680657899180, 15.67646871583599129870565356194, 16.58500952416513560215035175904, 17.64585867463752925860564590673, 18.5338517643056920005190509363, 19.26636195082424925961135676421, 20.1610706872556802364252008254, 21.33207966718102907793895209543, 21.920906584827522779188816395926, 23.27609166587619148533967104320, 24.58367505356732290256484431418, 25.06190334585967295148884751460, 26.29391489183612911420475392197

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