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Properties

Label 1-975-975.458-r0-0-0
Degree $1$
Conductor $975$
Sign $0.574 + 0.818i$
Analytic cond. $4.52788$
Root an. cond. $4.52788$
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 + (−0.809 + 0.587i)14-s + (0.913 − 0.406i)16-s + (0.743 + 0.669i)17-s + (−0.669 + 0.743i)19-s + (0.207 + 0.978i)22-s + (−0.994 + 0.104i)23-s + (−0.743 + 0.669i)28-s + (0.669 + 0.743i)29-s + (0.309 + 0.951i)31-s + (0.866 − 0.5i)32-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 + (−0.809 + 0.587i)14-s + (0.913 − 0.406i)16-s + (0.743 + 0.669i)17-s + (−0.669 + 0.743i)19-s + (0.207 + 0.978i)22-s + (−0.994 + 0.104i)23-s + (−0.743 + 0.669i)28-s + (0.669 + 0.743i)29-s + (0.309 + 0.951i)31-s + (0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(4.52788\)
Root analytic conductor: \(4.52788\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (458, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 975,\ (0:\ ),\ 0.574 + 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.238910606 + 1.164692946i\)
\(L(\frac12)\) \(\approx\) \(2.238910606 + 1.164692946i\)
\(L(1)\) \(\approx\) \(1.770222388 + 0.2753105871i\)
\(L(1)\) \(\approx\) \(1.770222388 + 0.2753105871i\)

Euler product

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

−21.86761011590896811336402557131, −20.861777258950760848741388732011, −20.24315787747558031268290811894, −19.31307795178782993657295690954, −18.80928388372336872465595326137, −17.32179584961941394966272128110, −16.72272503925852153445962355841, −15.922301291178790651215839074153, −15.38777550892653621595503633036, −14.12569599942860300403008806911, −13.7722907358124012790329728868, −12.96910264102980859790360516136, −12.10636862420085528862851414235, −11.3444841688449025061657266839, −10.45043564222518101352139104315, −9.64279499775667933635530172355, −8.374874340361507778673528841015, −7.51186838498390894742113275545, −6.49840482469959191284079603466, −6.02597410808434328513471338199, −4.9150475054183622926812110598, −3.95228899333521148789568015177, −3.20111430352336826239518533606, −2.32271460544447762629239763747, −0.76182429681803701451629496483, 1.52689456777066097996756518733, 2.429252275256840434935429061287, 3.46229475953890856801936247004, 4.19582317932480749630505463114, 5.29570397618732089319575477637, 6.07875065244391215380033541875, 6.811195173233782734583299416919, 7.75055342394122332862362441107, 8.87653553323993020251584741602, 10.12467058862485335882719399265, 10.4016919167967299588439188203, 11.82036204751407214344884357765, 12.44433246067978463867325794031, 12.79841756115017392788140395276, 14.00473317791162680714152062290, 14.60627870344229699516622577964, 15.467287252510129238409601708232, 16.08415277395258149001975302208, 16.9036134937301907059282108415, 17.92026999913075997624245043822, 19.03619923059459429845259219547, 19.56749725726302986782075073669, 20.39553375685723342017015880870, 21.209577061146902368295367689099, 21.92483114787966038482620237559

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