Properties

Label 1-99-99.38-r1-0-0
Degree $1$
Conductor $99$
Sign $0.578 - 0.815i$
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  + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.913 + 0.406i)5-s + (−0.978 + 0.207i)7-s + (−0.309 − 0.951i)8-s + 10-s + (−0.104 − 0.994i)13-s + (0.978 + 0.207i)14-s + (−0.104 + 0.994i)16-s + (0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.913 − 0.406i)20-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.913 + 0.406i)5-s + (−0.978 + 0.207i)7-s + (−0.309 − 0.951i)8-s + 10-s + (−0.104 − 0.994i)13-s + (0.978 + 0.207i)14-s + (−0.104 + 0.994i)16-s + (0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s + (−0.913 − 0.406i)20-s + (0.5 − 0.866i)23-s + (0.669 − 0.743i)25-s + (−0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(10.6390\)
Root analytic conductor: \(10.6390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 99,\ (1:\ ),\ 0.578 - 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6122694690 - 0.3161635852i\)
\(L(\frac12)\) \(\approx\) \(0.6122694690 - 0.3161635852i\)
\(L(1)\) \(\approx\) \(0.5768824020 - 0.09460237375i\)
\(L(1)\) \(\approx\) \(0.5768824020 - 0.09460237375i\)

Euler product

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

−29.50384083836446040412934575129, −28.704297747916902309324049952309, −27.72607924383962655571350418955, −26.784264142132990522595479355829, −25.9207348117363486271846533295, −24.84665490428344321846439069973, −23.69722678501548867630301524014, −23.032880569482877794144015393089, −21.30483475840892450759864474406, −19.864060675259481062262133910926, −19.435311914381555070400078920282, −18.32386412299686046979109826239, −16.85273390103764884416123532747, −16.17388276700198196172522278545, −15.29921083044685699484884347577, −13.83414220545563221396393352969, −12.22317332660813454333170896808, −11.21004093189309122037682446186, −9.75588387752761666343846998404, −8.87440631621142962700186093735, −7.51063117126112503214776643961, −6.63055851144842764317919089834, −4.93131162947237142854119982078, −3.11987353217343080422767450243, −0.9455892516800216984267882008, 0.56496239421396663821170223177, 2.76617356100218558015762897176, 3.763014758900906442338302924418, 6.12862035130248689487175697490, 7.45525306465736327017619772172, 8.39177857295027126561127567921, 9.85528463304591693740603278355, 10.71468243711158070126181842250, 12.07349601066620018342649692192, 12.80515240793507262096448812352, 14.827717822704560556464086640109, 15.89050509070542439265432169695, 16.742080356119349810866235800117, 18.183447820228128343244090130445, 19.072424267549644876612832166017, 19.77966217199194654968281988540, 20.89977305915120792350317724960, 22.29895827379357775200522092185, 23.107397136854045605558265996848, 24.717526038840909336452563995870, 25.677720883907303287774536560468, 26.6408451428212735007097545113, 27.50153548723594445216267135991, 28.42505127470613203419887422353, 29.5127036281674695463858343524

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