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Properties

Label 1-211-211.122-r0-0-0
Degree $1$
Conductor $211$
Sign $0.768 - 0.640i$
Analytic cond. $0.979879$
Root an. cond. $0.979879$
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  + (0.983 − 0.178i)2-s + (0.134 − 0.990i)3-s + (0.936 − 0.351i)4-s + (0.134 + 0.990i)5-s + (−0.0448 − 0.998i)6-s + (0.983 + 0.178i)7-s + (0.858 − 0.512i)8-s + (−0.963 − 0.266i)9-s + (0.309 + 0.951i)10-s + (−0.691 + 0.722i)11-s + (−0.222 − 0.974i)12-s + (0.753 − 0.657i)13-s + 14-s + 15-s + (0.753 − 0.657i)16-s + (−0.393 + 0.919i)17-s + ⋯
L(s)  = 1  + (0.983 − 0.178i)2-s + (0.134 − 0.990i)3-s + (0.936 − 0.351i)4-s + (0.134 + 0.990i)5-s + (−0.0448 − 0.998i)6-s + (0.983 + 0.178i)7-s + (0.858 − 0.512i)8-s + (−0.963 − 0.266i)9-s + (0.309 + 0.951i)10-s + (−0.691 + 0.722i)11-s + (−0.222 − 0.974i)12-s + (0.753 − 0.657i)13-s + 14-s + 15-s + (0.753 − 0.657i)16-s + (−0.393 + 0.919i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $0.768 - 0.640i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ 0.768 - 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.198635075 - 0.7961035966i\)
\(L(\frac12)\) \(\approx\) \(2.198635075 - 0.7961035966i\)
\(L(1)\) \(\approx\) \(1.907828209 - 0.5060777642i\)
\(L(1)\) \(\approx\) \(1.907828209 - 0.5060777642i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (0.983 - 0.178i)T \)
3 \( 1 + (0.134 - 0.990i)T \)
5 \( 1 + (0.134 + 0.990i)T \)
7 \( 1 + (0.983 + 0.178i)T \)
11 \( 1 + (-0.691 + 0.722i)T \)
13 \( 1 + (0.753 - 0.657i)T \)
17 \( 1 + (-0.393 + 0.919i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.753 - 0.657i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + (-0.691 - 0.722i)T \)
41 \( 1 + (-0.995 - 0.0896i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.963 + 0.266i)T \)
53 \( 1 + (0.936 + 0.351i)T \)
59 \( 1 + (-0.995 - 0.0896i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (0.858 + 0.512i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.473 + 0.880i)T \)
97 \( 1 + (-0.550 + 0.834i)T \)
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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

−26.75085045134371310190159711014, −25.72161159614507469993140849683, −24.767992899008672052666561813881, −23.868911679643578438172181604774, −23.174638777311103420119544813354, −21.810906646563174535868035259200, −21.12760621117545746061046933330, −20.692910592812118383438822244364, −19.77450841610022887223532259307, −18.01476562444368930234503085375, −16.62736388648182975407933931677, −16.282383860189520153757454202697, −15.28411534114075708720692465165, −14.09543838916991880339220031925, −13.60478623442790606299747367148, −12.17073328487768413694072074456, −11.23793417843546855600434771496, −10.35093369217421573439597083861, −8.68066801392602981025043570093, −8.131571108780671173831295939016, −6.29626081928086670356703854779, −5.07104796218012177739379978234, −4.609313897317092985464636722320, −3.41242662007029937024279750564, −1.86127967443145806871367711792, 1.81329268620967722413202766501, 2.51001383504660676728385081594, 3.91587787817070342791991401920, 5.46166506686785883961290665320, 6.35411576315615762603289414556, 7.403187056838476464382941798613, 8.28144605000258472803190734206, 10.35127268254045413594458041073, 11.1259433379562433002983595240, 12.07427582232578942240134519485, 13.17530820146468664326154900257, 13.84677413777440650646517457096, 14.975823193284596711553965458925, 15.38264354181312922425057481048, 17.38315354109939696178699138097, 18.06159295882321345120893204241, 19.06880477406782185735477023047, 20.05749583620317198088603146682, 21.0196745879644283947093394189, 21.96286017709588641194387396219, 23.07492410678629260252428174384, 23.60628921539079251350718375462, 24.4970581736063420184672348793, 25.5822760517136641157106204550, 26.05538709296128123751458446375

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