Properties

Label 1-99-99.2-r0-0-0
Degree $1$
Conductor $99$
Sign $-0.954 + 0.296i$
Analytic cond. $0.459754$
Root an. cond. $0.459754$
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

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

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.954 + 0.296i$
Analytic conductor: \(0.459754\)
Root analytic conductor: \(0.459754\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (0:\ ),\ -0.954 + 0.296i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1027327911 + 0.6769040741i\)
\(L(\frac12)\) \(\approx\) \(0.1027327911 + 0.6769040741i\)
\(L(1)\) \(\approx\) \(0.5355217282 + 0.5733696634i\)
\(L(1)\) \(\approx\) \(0.5355217282 + 0.5733696634i\)

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

−29.28901000697245068676752131670, −28.79341178358813149385972628352, −27.56740883832876472235057278804, −26.75955944474319560852697185068, −25.524889923716495272318165192454, −24.16562949362208223488144451077, −23.09742333522216095614927508544, −22.03703000603801219520174799989, −20.973062008878554788576938054436, −19.860033282008740590748588453094, −19.50794430768432456828633257688, −17.7323492083498155869267563230, −17.06127049654748668739098303701, −15.73364444329952672509868261234, −13.9146409841631858324621277834, −13.06855715467739282096163408102, −12.19559521047561540123830589484, −10.83061002634953723429738366506, −9.66522108114257122094772151100, −8.80514116963747808979347241388, −7.25402281985945165578674340329, −5.21786736602733331456763481974, −4.15195702566572105970375016469, −2.58543558663710666161186632005, −0.7356281352931826759879323811, 2.61693545948309226030581450456, 4.33171969394337556628597711032, 6.0187149684587459981610691531, 6.70527854860920907109734931978, 8.07895933807771964512171448348, 9.4083079206820185381408015005, 10.39976964429902517995820463475, 12.14738795960630692470354995833, 13.431582564642454863093158772460, 14.67961772941120919130750149750, 15.30125229456478223020397282293, 16.55609302268999328928452017038, 17.66294809341358554915654561622, 18.77075045519516981941551537137, 19.37601875120588994178669261276, 21.496177106512396877335063560254, 22.35008447465764263518990243882, 23.08187053774111477615039858864, 24.47381748944896630806548913082, 25.283343142210717317950341823515, 26.3324030590565563459093777785, 26.92870706734892090913698992651, 28.30185830775350070831282636439, 29.319008678701464933376351393141, 30.77181227824400208197677695101

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