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Properties

Label 1-211-211.210-r1-0-0
Degree $1$
Conductor $211$
Sign $1$
Analytic cond. $22.6750$
Root an. cond. $22.6750$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s + 20-s + 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s + 20-s + 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $1$
Analytic conductor: \(22.6750\)
Root analytic conductor: \(22.6750\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{211} (210, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9481851439\)
\(L(\frac12)\) \(\approx\) \(0.9481851439\)
\(L(1)\) \(\approx\) \(0.6488284725\)
\(L(1)\) \(\approx\) \(0.6488284725\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−26.44812730185330956266702364066, −25.62227070947778908964164204072, −24.76335796247969651539923113975, −23.86114873425633650045632429408, −22.38168104392155683608635465276, −21.97907890564437122192884621682, −20.665377865497271854631650925546, −19.72967155343978157714447896796, −18.41365890619832495328687145277, −18.01256211099126690087618199597, −16.87295457298272750517829750225, −16.372217398140040559861306446566, −15.361471334959767254290074974641, −13.71269109560770303691787283577, −12.646337939042138274444347502056, −11.53887757850298224039848773733, −10.62654708164555372351052483818, −9.63099900073932283027650980744, −9.01895091518082436280415387152, −7.19583608303278313837696209617, −6.30600183193249909735195286970, −5.735486160766875952822439509496, −3.73163529371871506583760249930, −1.98603116575114431707792575129, −0.787457686946212492015908362229, 0.787457686946212492015908362229, 1.98603116575114431707792575129, 3.73163529371871506583760249930, 5.735486160766875952822439509496, 6.30600183193249909735195286970, 7.19583608303278313837696209617, 9.01895091518082436280415387152, 9.63099900073932283027650980744, 10.62654708164555372351052483818, 11.53887757850298224039848773733, 12.646337939042138274444347502056, 13.71269109560770303691787283577, 15.361471334959767254290074974641, 16.372217398140040559861306446566, 16.87295457298272750517829750225, 18.01256211099126690087618199597, 18.41365890619832495328687145277, 19.72967155343978157714447896796, 20.665377865497271854631650925546, 21.97907890564437122192884621682, 22.38168104392155683608635465276, 23.86114873425633650045632429408, 24.76335796247969651539923113975, 25.62227070947778908964164204072, 26.44812730185330956266702364066

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