Properties

Label 1-99-99.43-r1-0-0
Degree $1$
Conductor $99$
Sign $-0.939 - 0.342i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 8-s − 10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 17-s − 19-s + (−0.5 − 0.866i)20-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 8-s − 10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 17-s − 19-s + (−0.5 − 0.866i)20-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(10.6390\)
Root analytic conductor: \(10.6390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (1:\ ),\ -0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2320565332 + 1.316057997i\)
\(L(\frac12)\) \(\approx\) \(-0.2320565332 + 1.316057997i\)
\(L(1)\) \(\approx\) \(0.7030567709 + 0.8378704325i\)
\(L(1)\) \(\approx\) \(0.7030567709 + 0.8378704325i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−29.13513704502863227606308156735, −28.28828723841218291268182397409, −27.375964820771569561355720483830, −26.378879295572039115998929554718, −24.51661724080266868780618246638, −23.78911483416642691363194694506, −23.0088962070407790288686011229, −21.59199034838795228927460317929, −20.65394588419516048952943867295, −19.95345970400574299517484647437, −18.89563508072086824429076616278, −17.5149683098857924510119945261, −16.30218906855167951806245280671, −14.93545916594799116929495707542, −13.72842859379038253759388272365, −12.852471662013827480753092522465, −11.61114223896028114724131726138, −10.78516402767088491990454892612, −9.31262965043258123555196676826, −8.162133521255278847591233721719, −6.33045376215435625326442922112, −4.55678953772056721561370949349, −4.078387902583025224995246199110, −1.98264787135389382957847533985, −0.49788782127968831009992032933, 2.66233725267163712183441952150, 4.0430384737051644701843642871, 5.51828678000151764654658128687, 6.6460804957658615088295948707, 7.90263254102433317574853118837, 8.86420594182160525874561103042, 10.7758378771751708528416791984, 11.94136037250482192639477705710, 13.142292992796741140452942532160, 14.49226787911510944669500416921, 15.239529176268795551268304869063, 16.03378348342793421014649525132, 17.73912771477169735369899150496, 18.23987379755836539761518778970, 19.707959964238709025147895940227, 21.327105967759618964136973795519, 22.10262789262436026729342848600, 23.10002486277109135678581621838, 24.00481679828754298678041869275, 25.179760217911413720723878453215, 25.92589082784004855038372473728, 27.15214284241652863830755486331, 27.83541967439617885456745399337, 29.624302452829281049338010696931, 30.704853365679907779928695423628

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