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Properties

Label 1-15e2-225.202-r1-0-0
Degree $1$
Conductor $225$
Sign $-0.658 + 0.752i$
Analytic cond. $24.1796$
Root an. cond. $24.1796$
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  + (−0.743 + 0.669i)2-s + (0.104 − 0.994i)4-s + (−0.866 − 0.5i)7-s + (0.587 + 0.809i)8-s + (0.669 + 0.743i)11-s + (−0.743 − 0.669i)13-s + (0.978 − 0.207i)14-s + (−0.978 − 0.207i)16-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.994 − 0.104i)22-s + (0.207 + 0.978i)23-s + 26-s + (−0.587 + 0.809i)28-s + (−0.913 + 0.406i)29-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (0.104 − 0.994i)4-s + (−0.866 − 0.5i)7-s + (0.587 + 0.809i)8-s + (0.669 + 0.743i)11-s + (−0.743 − 0.669i)13-s + (0.978 − 0.207i)14-s + (−0.978 − 0.207i)16-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.994 − 0.104i)22-s + (0.207 + 0.978i)23-s + 26-s + (−0.587 + 0.809i)28-s + (−0.913 + 0.406i)29-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(24.1796\)
Root analytic conductor: \(24.1796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (1:\ ),\ -0.658 + 0.752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2424827255 + 0.5345502014i\)
\(L(\frac12)\) \(\approx\) \(0.2424827255 + 0.5345502014i\)
\(L(1)\) \(\approx\) \(0.5960680035 + 0.1806333101i\)
\(L(1)\) \(\approx\) \(0.5960680035 + 0.1806333101i\)

Euler product

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

−26.33894367020821268786747895599, −24.97271463226965572412485186129, −24.37388139283616443431922478891, −22.61344424531217499885532553980, −22.08535425394474193485564115506, −21.16205597359171193090423950639, −20.033192092664991769052860845474, −19.148130029005472076922690319507, −18.70537035506626875235415944396, −17.32756448694388827193653261229, −16.60969166734909669676250291643, −15.69136292183019708890256793647, −14.27630610471807045148114268541, −13.05468843993002732359015301549, −12.1704341359772819063195197998, −11.3268678646535063472822179825, −10.10943756864440509520852944262, −9.258611946804285531881954551297, −8.43048070353401792901402676421, −7.071023567443409073012950530286, −6.03845887485883448155434888084, −4.20608035516686849040663512430, −3.11191425398995312133137776276, −1.90713485903110140283367043527, −0.28285072397100387525782264847, 1.10647383850070566148299171474, 2.81166402885678546164657394777, 4.51103632494786657000799811845, 5.71463212989568537665964076440, 7.04349504417520157807321675221, 7.429454183624301450257964835183, 9.1172956342808713434895889510, 9.65636540930191177213770487014, 10.70402375133953353485890347688, 11.96975384816893654262502228102, 13.30123129629971979104020468449, 14.270830591007068356849988725439, 15.41592010856528127743807916580, 16.06815532959603237262612339888, 17.297101756940020692808584526093, 17.70659596059601074205140665823, 19.0857105165542015156419508967, 19.81270455864868551969867450038, 20.49709325477985659442426000911, 22.388519925144573804658757599742, 22.743331261108742099642924696449, 23.9986662263175268644407714817, 24.85952732555435954386365571030, 25.64767236999798827137712696166, 26.50228938571656129123377461403

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