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Properties

Label 1-99-99.94-r1-0-0
Degree $1$
Conductor $99$
Sign $0.296 - 0.954i$
Analytic cond. $10.6390$
Root an. cond. $10.6390$
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  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.104 + 0.994i)5-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s − 10-s + (−0.913 − 0.406i)13-s + (0.669 − 0.743i)14-s + (0.913 − 0.406i)16-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (−0.104 − 0.994i)20-s + (−0.5 − 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.309 − 0.951i)26-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.104 + 0.994i)5-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s − 10-s + (−0.913 − 0.406i)13-s + (0.669 − 0.743i)14-s + (0.913 − 0.406i)16-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (−0.104 − 0.994i)20-s + (−0.5 − 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.309 − 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.296 - 0.954i$
Analytic conductor: \(10.6390\)
Root analytic conductor: \(10.6390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (1:\ ),\ 0.296 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2355640385 - 0.1734834668i\)
\(L(\frac12)\) \(\approx\) \(0.2355640385 - 0.1734834668i\)
\(L(1)\) \(\approx\) \(0.6355331943 + 0.3092955316i\)
\(L(1)\) \(\approx\) \(0.6355331943 + 0.3092955316i\)

Euler product

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

−29.5836491883234055440289324471, −29.071240130104564537786693154494, −27.96288184739649309621006117998, −27.33967804175928394791682959528, −25.86065462960693485711471847185, −24.66078186882191030493802826993, −23.533333640254929264526345379987, −22.41682475112391984162613502764, −21.4386451598623258885293821736, −20.51369002720823583041741595855, −19.44725085005870626549239518069, −18.67089629528942727319938131274, −17.25017417849306268379846627546, −16.17011113278204225588283560068, −14.68934068121204344821114450096, −13.37195180676148847565805548864, −12.32649258377635330403695074267, −11.76974983413793651593517467035, −9.930406737413477562449580976779, −9.26482246188996720005356017838, −7.97781961250572526437419842861, −5.80052629114344919030964273590, −4.68969484904433095447883344021, −3.22157225253061690379801663636, −1.66910827511000909401424042793, 0.124522044588925134355987689496, 3.03937242581818596294809746711, 4.352001594175097078929798016196, 6.029285686475049446157826073458, 7.01177604325322829435942043928, 7.938286135399354118291044914126, 9.643818659468679680899494045651, 10.5526437448217451589718163304, 12.37734153949050987282679188431, 13.57842135249378029466269470603, 14.58187887277735151864538450877, 15.46519769506742951794830479351, 16.719465781851607219031683320717, 17.59658962972232127619950764196, 18.82244126724806026825487824128, 19.73063379658382536917765031085, 21.56230769118084972588489930382, 22.53276375903058496712677032317, 23.206869151753660604101751004944, 24.30921160163818753115609179432, 25.56518380931081953553094689505, 26.3463427746907266156547200384, 26.99273952739150634378040429949, 28.31116140607085728768639346696, 29.8977707547454838151355415418

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