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Properties

Label 1-99-99.59-r1-0-0
Degree $1$
Conductor $99$
Sign $0.999 + 0.0111i$
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  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (0.978 − 0.207i)5-s + (−0.104 − 0.994i)7-s + (0.809 + 0.587i)8-s + 10-s + (0.669 − 0.743i)13-s + (0.104 − 0.994i)14-s + (0.669 + 0.743i)16-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)19-s + (0.978 + 0.207i)20-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (0.809 − 0.587i)26-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (0.978 − 0.207i)5-s + (−0.104 − 0.994i)7-s + (0.809 + 0.587i)8-s + 10-s + (0.669 − 0.743i)13-s + (0.104 − 0.994i)14-s + (0.669 + 0.743i)16-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)19-s + (0.978 + 0.207i)20-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)25-s + (0.809 − 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.999 + 0.0111i$
Analytic conductor: \(10.6390\)
Root analytic conductor: \(10.6390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (1:\ ),\ 0.999 + 0.0111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.685802941 + 0.02051661415i\)
\(L(\frac12)\) \(\approx\) \(3.685802941 + 0.02051661415i\)
\(L(1)\) \(\approx\) \(2.249366739 + 0.05610601411i\)
\(L(1)\) \(\approx\) \(2.249366739 + 0.05610601411i\)

Euler product

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

−29.7941548601872562958796977885, −28.852898725193074491587963382934, −28.18146815185408221173775930179, −26.38736329726148475602012955661, −25.12974590391759873935761551771, −24.77335336297141374274072134308, −23.25622963952359734802720259251, −22.34291715932501956402954854393, −21.34194768388916896848651754565, −20.80021696708394487442402566125, −19.17136221358923375501847757069, −18.27146455655480068915227546699, −16.71596463542801669489427064249, −15.57498156557796617167187754613, −14.45231573572195660642757048998, −13.553195840776865458065363573904, −12.45940200661337955780537671413, −11.33309940511468143041096900465, −10.08748037238663612348454265714, −8.78549769167699739438726594166, −6.72350252836588250893375135064, −5.91496012555416373785750630733, −4.66720816092786048160114886669, −2.91241588637349249487787090820, −1.82032080355955512539737380009, 1.537819220715635234215962542965, 3.23878812981664456107081630998, 4.62068151768167578564977916437, 5.91163391345570644842487536540, 6.920425434986449551039122430410, 8.42526228320903921884982527643, 10.20429770394862653137432890514, 11.106168655869877009122371871614, 12.90629486801039128797502741282, 13.33359542008291754446321718229, 14.46874236792095060594225982351, 15.66199084573634284775680671807, 16.95259388558132252401123056984, 17.59868319527231657027817914835, 19.52837304822510356884124610725, 20.58495774105885631602537411571, 21.38320041451543379904942382541, 22.454725930214168374469773395361, 23.490540130407806872743529286304, 24.36892624277416967816700159388, 25.61541068227311421264762268575, 26.10116204831624847597407627706, 27.832271011740416246216619933547, 29.12456893968557375918316114303, 29.84685902416990705137912110732

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