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Properties

Label 1-201-201.152-r0-0-0
Degree $1$
Conductor $201$
Sign $0.860 - 0.509i$
Analytic cond. $0.933440$
Root an. cond. $0.933440$
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  + (−0.327 − 0.945i)2-s + (−0.786 + 0.618i)4-s + (−0.959 + 0.281i)5-s + (−0.981 − 0.189i)7-s + (0.841 + 0.540i)8-s + (0.580 + 0.814i)10-s + (0.723 + 0.690i)11-s + (−0.0475 − 0.998i)13-s + (0.142 + 0.989i)14-s + (0.235 − 0.971i)16-s + (0.786 + 0.618i)17-s + (0.981 − 0.189i)19-s + (0.580 − 0.814i)20-s + (0.415 − 0.909i)22-s + (0.995 + 0.0950i)23-s + ⋯
L(s)  = 1  + (−0.327 − 0.945i)2-s + (−0.786 + 0.618i)4-s + (−0.959 + 0.281i)5-s + (−0.981 − 0.189i)7-s + (0.841 + 0.540i)8-s + (0.580 + 0.814i)10-s + (0.723 + 0.690i)11-s + (−0.0475 − 0.998i)13-s + (0.142 + 0.989i)14-s + (0.235 − 0.971i)16-s + (0.786 + 0.618i)17-s + (0.981 − 0.189i)19-s + (0.580 − 0.814i)20-s + (0.415 − 0.909i)22-s + (0.995 + 0.0950i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.860 - 0.509i$
Analytic conductor: \(0.933440\)
Root analytic conductor: \(0.933440\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 201,\ (0:\ ),\ 0.860 - 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6861661700 - 0.1879097323i\)
\(L(\frac12)\) \(\approx\) \(0.6861661700 - 0.1879097323i\)
\(L(1)\) \(\approx\) \(0.6890845330 - 0.2117922335i\)
\(L(1)\) \(\approx\) \(0.6890845330 - 0.2117922335i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.327 - 0.945i)T \)
5 \( 1 + (-0.959 + 0.281i)T \)
7 \( 1 + (-0.981 - 0.189i)T \)
11 \( 1 + (0.723 + 0.690i)T \)
13 \( 1 + (-0.0475 - 0.998i)T \)
17 \( 1 + (0.786 + 0.618i)T \)
19 \( 1 + (0.981 - 0.189i)T \)
23 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.0475 + 0.998i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.928 - 0.371i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.580 + 0.814i)T \)
53 \( 1 + (-0.142 - 0.989i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (-0.723 + 0.690i)T \)
71 \( 1 + (0.786 - 0.618i)T \)
73 \( 1 + (0.723 - 0.690i)T \)
79 \( 1 + (0.888 + 0.458i)T \)
83 \( 1 + (-0.235 + 0.971i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
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

−26.77029500903541205804148045047, −26.19104967405170773622131373812, −24.854727165961474044222163392746, −24.424169397763457303258681787803, −23.089923917717431574102453248789, −22.77242979288644117628290645728, −21.44543223759541608025499924349, −19.897308805641126215487674671320, −19.11770201149862442751859370399, −18.56461022866483207798831279467, −16.92299925670280869232518487201, −16.36596737311518898615083010460, −15.64955038019194399090665696964, −14.49713965236583870923324736562, −13.55472779012573441856801920572, −12.289132731729404610855872350338, −11.24951770897267774424338475451, −9.618102835190341699710183086028, −9.02563073803255924063765934553, −7.789262172488796041342066027287, −6.85970325033470384489894158489, −5.80410332819023900792398204544, −4.44572824968907468390267067510, −3.32984648363124685922723296953, −0.852328868998141762548229105308, 1.063903175412626301543511179, 3.048913334519195663257033522006, 3.60651736422466476476050641178, 5.004757539520323094077243746744, 6.8633447403954534754044085632, 7.82665178521767035961344027539, 9.052084976181463989364137829143, 10.090904901547922039365651209829, 10.936655308996589671409337962479, 12.23252035871167317610857737582, 12.618449480595488798724146075024, 14.00939433267706645958820104371, 15.20957360224717680666109997364, 16.32342080851456739384924797885, 17.36914820560608867964077148621, 18.42685349199792981135892130372, 19.52152058429279098787148522862, 19.79949454301628176973054750109, 20.88046243797902440718553831251, 22.345853152997635381949628417292, 22.65751434917187036601200034511, 23.60720156850270257404038400770, 25.25234596595536266732325597506, 26.02061902251433245453645094377, 27.127278690571804733885801000056

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