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Properties

Label 1-211-211.18-r1-0-0
Degree $1$
Conductor $211$
Sign $-0.639 + 0.769i$
Analytic cond. $22.6750$
Root an. cond. $22.6750$
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  + (−0.858 + 0.512i)2-s + (0.393 − 0.919i)3-s + (0.473 − 0.880i)4-s + (−0.393 − 0.919i)5-s + (0.134 + 0.990i)6-s + (−0.858 − 0.512i)7-s + (0.0448 + 0.998i)8-s + (−0.691 − 0.722i)9-s + (0.809 + 0.587i)10-s + (0.753 + 0.657i)11-s + (−0.623 − 0.781i)12-s + (−0.550 − 0.834i)13-s + 14-s − 15-s + (−0.550 − 0.834i)16-s + (−0.936 + 0.351i)17-s + ⋯
L(s)  = 1  + (−0.858 + 0.512i)2-s + (0.393 − 0.919i)3-s + (0.473 − 0.880i)4-s + (−0.393 − 0.919i)5-s + (0.134 + 0.990i)6-s + (−0.858 − 0.512i)7-s + (0.0448 + 0.998i)8-s + (−0.691 − 0.722i)9-s + (0.809 + 0.587i)10-s + (0.753 + 0.657i)11-s + (−0.623 − 0.781i)12-s + (−0.550 − 0.834i)13-s + 14-s − 15-s + (−0.550 − 0.834i)16-s + (−0.936 + 0.351i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.639 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.639 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $-0.639 + 0.769i$
Analytic conductor: \(22.6750\)
Root analytic conductor: \(22.6750\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (1:\ ),\ -0.639 + 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1293686066 - 0.2757308026i\)
\(L(\frac12)\) \(\approx\) \(-0.1293686066 - 0.2757308026i\)
\(L(1)\) \(\approx\) \(0.5205722297 - 0.2590886054i\)
\(L(1)\) \(\approx\) \(0.5205722297 - 0.2590886054i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (-0.858 + 0.512i)T \)
3 \( 1 + (0.393 - 0.919i)T \)
5 \( 1 + (-0.393 - 0.919i)T \)
7 \( 1 + (-0.858 - 0.512i)T \)
11 \( 1 + (0.753 + 0.657i)T \)
13 \( 1 + (-0.550 - 0.834i)T \)
17 \( 1 + (-0.936 + 0.351i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (0.550 + 0.834i)T \)
31 \( 1 + (-0.623 - 0.781i)T \)
37 \( 1 + (0.753 - 0.657i)T \)
41 \( 1 + (0.963 + 0.266i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (-0.691 + 0.722i)T \)
53 \( 1 + (0.473 + 0.880i)T \)
59 \( 1 + (-0.963 - 0.266i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (-0.0448 + 0.998i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.995 + 0.0896i)T \)
97 \( 1 + (-0.983 - 0.178i)T \)
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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

−26.989509747043450182209984519710, −26.41395653633840136545863485192, −25.56675255072926950662590590908, −24.69305066522848707690269215833, −22.86968809146272145645222469942, −21.91766616511082826833392283955, −21.641940690519309492629004721246, −20.18584562325155345663738655645, −19.36831303911831162748368492479, −18.9178317269903140812817387650, −17.60093562714869033645075401266, −16.35731879245255227496413241413, −15.85588448619206382409848279910, −14.71634228632961515506281941561, −13.613560704881018032593993099919, −11.9321893750940842238010353763, −11.33191021566311068097675513995, −10.1557280206058011989924132974, −9.43566383276290632729047296718, −8.55389777268175872535031171266, −7.25244916475997881687438482064, −6.12569367577935103274652891991, −4.06177191973715810902914092699, −3.26020247165827760786206709309, −2.23992427432058074451426627319, 0.14327478739437591161431168390, 1.11596432551185066869959401095, 2.60062699950596792445059736583, 4.422509245045764373496332210897, 6.02304604968013418766761754973, 7.01513938158448398586713555111, 7.79285440257283653574335814092, 8.926760734188311603710703458734, 9.581538703697720072006063152012, 11.07069554360133888700265623336, 12.3966029419093522029340496117, 13.08508729440220839308043092115, 14.399720922249293326702875923757, 15.4018226962331462064407848628, 16.457182363489329838952997496264, 17.35941548742646484400947406005, 18.06022805657490131677426936841, 19.54625888850355513593318559889, 19.79081633337935356790769294587, 20.40889330624455827494497151255, 22.46491970215065404182160871295, 23.39964959865180544916051124803, 24.3412006663370006201395765612, 24.84419608440332962337062289316, 25.812241472757085464938948620957

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