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Properties

Label 1-15e2-225.128-r0-0-0
Degree $1$
Conductor $225$
Sign $0.999 + 0.0279i$
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  + (−0.994 − 0.104i)2-s + (0.978 + 0.207i)4-s + (−0.866 + 0.5i)7-s + (−0.951 − 0.309i)8-s + (0.104 − 0.994i)11-s + (0.994 − 0.104i)13-s + (0.913 − 0.406i)14-s + (0.913 + 0.406i)16-s + (0.951 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (−0.207 + 0.978i)22-s + (−0.406 − 0.913i)23-s − 26-s + (−0.951 + 0.309i)28-s + (0.669 − 0.743i)29-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (0.978 + 0.207i)4-s + (−0.866 + 0.5i)7-s + (−0.951 − 0.309i)8-s + (0.104 − 0.994i)11-s + (0.994 − 0.104i)13-s + (0.913 − 0.406i)14-s + (0.913 + 0.406i)16-s + (0.951 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (−0.207 + 0.978i)22-s + (−0.406 − 0.913i)23-s − 26-s + (−0.951 + 0.309i)28-s + (0.669 − 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.999 + 0.0279i$
Analytic conductor: \(1.04489\)
Root analytic conductor: \(1.04489\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (0:\ ),\ 0.999 + 0.0279i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7367156721 + 0.01028715982i\)
\(L(\frac12)\) \(\approx\) \(0.7367156721 + 0.01028715982i\)
\(L(1)\) \(\approx\) \(0.7063697290 + 0.01339261071i\)
\(L(1)\) \(\approx\) \(0.7063697290 + 0.01339261071i\)

Euler product

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

−26.09511056592708235170984439199, −25.87796332436291486114744873222, −24.9369518756181409519534232165, −23.60677868113560020579770679701, −23.01701384178682483376739427688, −21.57963904139273185349907811085, −20.51596520161000842801996505496, −19.78853791820894935759308328021, −18.921114665339787663093966072686, −17.91115318463140638810947801308, −17.08047422535248750232518068055, −16.07312751226966280964278660292, −15.410360925650835003857648755675, −14.08443710490581941751649586612, −12.8486777873484538610031566559, −11.76996178687020913594701268192, −10.62443785566135226449195440413, −9.7733083154644738922117081470, −8.93131691445353500721258050714, −7.592490112020650996175581891772, −6.82065658531354012898216622895, −5.71412337339458724772179133031, −3.92905029342648668304421514964, −2.58196508477759633027870989516, −1.03252922763085897519025977194, 1.04682978604221400248189344780, 2.69327600199278089413921028851, 3.68669811296429015266636678571, 5.92068962873352515842533042491, 6.39989500695422591791135314260, 8.03696391625344549197678601636, 8.66616862844706706482466751917, 9.8513143341552158151117386655, 10.65416044374391246189479198018, 11.83994266816999737033586613840, 12.68313572789916306145795591740, 14.03060459010375109056184346091, 15.36415730213692936699268598417, 16.24860569527550690394523135156, 16.8205531700448048375847926340, 18.22527579186952985575449310019, 18.865140991239538943597467984871, 19.5578475308713046523191129746, 20.80639194452409714810456373642, 21.47931560314904325298742128820, 22.69705056335640307730476069061, 23.805624967919190925434070386967, 24.98446661896752673509020597681, 25.52950978911587961853131528956, 26.475868860604069588953874762438

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