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Properties

Label 1-213-213.125-r1-0-0
Degree $1$
Conductor $213$
Sign $0.808 + 0.588i$
Analytic cond. $22.8900$
Root an. cond. $22.8900$
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  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 + 0.587i)5-s + (0.309 + 0.951i)7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)10-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s − 20-s + (−0.809 + 0.587i)22-s − 23-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 + 0.587i)5-s + (0.309 + 0.951i)7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)10-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s − 20-s + (−0.809 + 0.587i)22-s − 23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(213\)    =    \(3 \cdot 71\)
Sign: $0.808 + 0.588i$
Analytic conductor: \(22.8900\)
Root analytic conductor: \(22.8900\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{213} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 213,\ (1:\ ),\ 0.808 + 0.588i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.416933393 + 0.4614696939i\)
\(L(\frac12)\) \(\approx\) \(1.416933393 + 0.4614696939i\)
\(L(1)\) \(\approx\) \(0.9942028395 - 0.09528641731i\)
\(L(1)\) \(\approx\) \(0.9942028395 - 0.09528641731i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
71 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 - T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
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

−26.18790753260867917864485369649, −25.473257375292073941015214644088, −24.43775945088252086545352101890, −23.77033931233166603381257206579, −22.86218113549998203271792427944, −21.71790328365909290729672111692, −20.55796219344468031495867795266, −19.770515154976353302548535384977, −18.2364848921250241200455437572, −17.70210840824117234911029543053, −16.763052737179896552977674562835, −16.027440199302484664887362717800, −14.80052317793189021699220014868, −13.65248214244229032559942045843, −13.36255110505801098938719311771, −11.657310295084245960286854006675, −10.12431758403741564668651668280, −9.547258024155068429543431459254, −8.382586455517875480636740139239, −7.24036338524664797196740842398, −6.37447887614682365646268210777, −4.961342422589105124326006285026, −4.33570765154697140972331367833, −1.96366884113276363619498405115, −0.59101885554758839693475417414, 1.387082689049843768868710026926, 2.56681545730865689484485674624, 3.503938663108259984070732607670, 5.27064833818708814382282559821, 6.149482830543235019742912502351, 8.0470013205943720174497631721, 8.72900400399839837407534860793, 10.07338066926163852234948894204, 10.677539157611255242330900758682, 11.80007764324026221211220017981, 12.82760854827198863966955761929, 13.78755251517643518224643274187, 14.7397925109497515085332509828, 16.09701451656436083742912169751, 17.42354091675795889811827316020, 18.19727220762406700342022570793, 18.74447825673980792444024231962, 19.89900337031558950041316420700, 21.035764175400015609629870505235, 21.73833552046292248330750717089, 22.27810246152837387995132422470, 23.52555554678122567660839150982, 24.9233708539111895301286680924, 25.66946809276563627538395537069, 26.69967033879174660181879849665

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