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Properties

Label 1-211-211.151-r0-0-0
Degree $1$
Conductor $211$
Sign $-0.984 + 0.176i$
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  + (−0.550 + 0.834i)2-s + (−0.0448 − 0.998i)3-s + (−0.393 − 0.919i)4-s + (−0.0448 + 0.998i)5-s + (0.858 + 0.512i)6-s + (−0.550 − 0.834i)7-s + (0.983 + 0.178i)8-s + (−0.995 + 0.0896i)9-s + (−0.809 − 0.587i)10-s + (−0.963 − 0.266i)11-s + (−0.900 + 0.433i)12-s + (−0.691 + 0.722i)13-s + 14-s + 15-s + (−0.691 + 0.722i)16-s + (0.134 + 0.990i)17-s + ⋯
L(s)  = 1  + (−0.550 + 0.834i)2-s + (−0.0448 − 0.998i)3-s + (−0.393 − 0.919i)4-s + (−0.0448 + 0.998i)5-s + (0.858 + 0.512i)6-s + (−0.550 − 0.834i)7-s + (0.983 + 0.178i)8-s + (−0.995 + 0.0896i)9-s + (−0.809 − 0.587i)10-s + (−0.963 − 0.266i)11-s + (−0.900 + 0.433i)12-s + (−0.691 + 0.722i)13-s + 14-s + 15-s + (−0.691 + 0.722i)16-s + (0.134 + 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $-0.984 + 0.176i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ -0.984 + 0.176i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01261795188 + 0.1415482457i\)
\(L(\frac12)\) \(\approx\) \(0.01261795188 + 0.1415482457i\)
\(L(1)\) \(\approx\) \(0.4850393114 + 0.09994866231i\)
\(L(1)\) \(\approx\) \(0.4850393114 + 0.09994866231i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (-0.550 + 0.834i)T \)
3 \( 1 + (-0.0448 - 0.998i)T \)
5 \( 1 + (-0.0448 + 0.998i)T \)
7 \( 1 + (-0.550 - 0.834i)T \)
11 \( 1 + (-0.963 - 0.266i)T \)
13 \( 1 + (-0.691 + 0.722i)T \)
17 \( 1 + (0.134 + 0.990i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.691 + 0.722i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (-0.963 + 0.266i)T \)
41 \( 1 + (0.473 - 0.880i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.995 - 0.0896i)T \)
53 \( 1 + (-0.393 + 0.919i)T \)
59 \( 1 + (0.473 - 0.880i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (0.983 - 0.178i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.936 - 0.351i)T \)
97 \( 1 + (0.753 - 0.657i)T \)
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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

−26.52685436346853399557131395047, −25.49603144912473311433479142310, −24.71069116741484738909392245338, −22.92346228470589966615154256225, −22.362537062560115093688824699302, −21.18854011466987157903140150427, −20.659463340523034990683371420923, −19.90587010481613631473282114565, −18.72008027954535797218715775797, −17.71582887566059977888045630180, −16.539383103002214707751774417660, −16.07001191231005694709861840931, −14.884082728484867064552400424673, −13.25226529166705031256431457493, −12.4163867364714127882415558251, −11.57822510866517466585297221105, −10.203856778172490712972309851361, −9.62099862210425758148009869863, −8.70508690935860870715565923386, −7.74524800917025264660918213513, −5.52763750598127800149007562319, −4.76543239207963856651891091307, −3.3763951670632806554949351886, −2.32167158056225664666490611226, −0.1210565277183089529994733963, 1.78682839795819472178281660877, 3.33291019641558016854240951076, 5.24351301838699273746500971779, 6.46545987231845628736207214124, 7.16601350044247588692203148541, 7.80098151188429031857875483209, 9.22281777553877077030486233398, 10.44691021295718772158171942866, 11.22674745863479490782053793653, 12.91566489490706650680639010633, 13.76589145717811548253040474520, 14.55988434733765903689065874022, 15.679152916506957282245390463479, 16.86844657973229483386138149274, 17.6190251922468933215868838550, 18.55844570450544450541234708900, 19.283292098742536007966805448783, 19.92115819425197091555186241146, 21.805192197201239540984529679935, 22.86046148272596314818857390590, 23.697150227322997363510433074308, 24.093312220733002776243596139394, 25.538266972605372900823153572065, 26.139765588411946002151321847313, 26.63216238850156697583944830565

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