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Properties

Label 1-229-229.106-r1-0-0
Degree $1$
Conductor $229$
Sign $-0.791 - 0.610i$
Analytic cond. $24.6094$
Root an. cond. $24.6094$
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  + (0.837 + 0.546i)2-s + (−0.879 − 0.475i)3-s + (0.401 + 0.915i)4-s + (0.0825 − 0.996i)5-s + (−0.475 − 0.879i)6-s + (0.324 − 0.945i)7-s + (−0.164 + 0.986i)8-s + (0.546 + 0.837i)9-s + (0.614 − 0.789i)10-s + (−0.245 − 0.969i)11-s + (0.0825 − 0.996i)12-s + (−0.996 − 0.0825i)13-s + (0.789 − 0.614i)14-s + (−0.546 + 0.837i)15-s + (−0.677 + 0.735i)16-s + (−0.0825 + 0.996i)17-s + ⋯
L(s)  = 1  + (0.837 + 0.546i)2-s + (−0.879 − 0.475i)3-s + (0.401 + 0.915i)4-s + (0.0825 − 0.996i)5-s + (−0.475 − 0.879i)6-s + (0.324 − 0.945i)7-s + (−0.164 + 0.986i)8-s + (0.546 + 0.837i)9-s + (0.614 − 0.789i)10-s + (−0.245 − 0.969i)11-s + (0.0825 − 0.996i)12-s + (−0.996 − 0.0825i)13-s + (0.789 − 0.614i)14-s + (−0.546 + 0.837i)15-s + (−0.677 + 0.735i)16-s + (−0.0825 + 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.791 - 0.610i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ -0.791 - 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2917682442 - 0.8560689580i\)
\(L(\frac12)\) \(\approx\) \(0.2917682442 - 0.8560689580i\)
\(L(1)\) \(\approx\) \(1.044927635 - 0.1488870970i\)
\(L(1)\) \(\approx\) \(1.044927635 - 0.1488870970i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.837 + 0.546i)T \)
3 \( 1 + (-0.879 - 0.475i)T \)
5 \( 1 + (0.0825 - 0.996i)T \)
7 \( 1 + (0.324 - 0.945i)T \)
11 \( 1 + (-0.245 - 0.969i)T \)
13 \( 1 + (-0.996 - 0.0825i)T \)
17 \( 1 + (-0.0825 + 0.996i)T \)
19 \( 1 + (-0.0825 - 0.996i)T \)
23 \( 1 + (0.614 + 0.789i)T \)
29 \( 1 + (-0.324 + 0.945i)T \)
31 \( 1 + (-0.969 + 0.245i)T \)
37 \( 1 + (-0.677 - 0.735i)T \)
41 \( 1 + (-0.837 - 0.546i)T \)
43 \( 1 + (-0.677 - 0.735i)T \)
47 \( 1 + (0.837 - 0.546i)T \)
53 \( 1 + (-0.879 - 0.475i)T \)
59 \( 1 + (-0.735 - 0.677i)T \)
61 \( 1 + (0.546 + 0.837i)T \)
67 \( 1 + (0.837 - 0.546i)T \)
71 \( 1 + (-0.245 + 0.969i)T \)
73 \( 1 + (0.164 - 0.986i)T \)
79 \( 1 + (-0.324 - 0.945i)T \)
83 \( 1 + (-0.677 + 0.735i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.986 - 0.164i)T \)
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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

−26.8375099195159255289671501104, −25.35012747619379140933671066861, −24.48671862630393273417333917289, −23.280102936590311629787066399674, −22.58824221682227134946737752950, −22.07428187819259178827246684485, −21.1740146009965982225753916585, −20.33249414224601445165883141272, −18.75549866414143079059629224661, −18.36185434458500335654424806502, −17.09901787472249758897972001662, −15.68676520223940125867773328597, −15.006872844204621222401485318196, −14.352177817486772641074347419897, −12.75140328724751185595533575246, −11.95240825664548047870280696613, −11.25273452133622124038942016775, −10.1556445563719011403440014017, −9.53825760837057655812954172562, −7.32787366197811859951141892576, −6.33171617477285138283756969232, −5.297631380100570863239301452107, −4.48143472217824736344734375329, −3.01076904241228220174933855081, −1.94101700839165145341264566308, 0.23108473359468059866978725996, 1.764328056968689917503096939692, 3.70557222449461314563240648723, 4.96372463817616265062521290767, 5.46142346346900078562503985900, 6.83289185096420511905691969398, 7.64273411229396794373821366696, 8.76751116965009796888215949101, 10.597139646516649617637883434964, 11.47037347101601971787831574807, 12.58052406556522536013047290854, 13.19712599823693890669760988907, 14.03827810605225929744938535346, 15.441987625525722924841493110941, 16.51714861656808419060649507906, 17.06284251623651978257521356190, 17.67895637253762720948351760815, 19.36529009086991504157107981043, 20.29762242758623614144700359356, 21.54311979184514213365458399800, 21.99513346941124244133126235711, 23.42880842334046338687342530068, 23.890190130836451754218433288523, 24.359450749950065733688428553306, 25.430837850679330452420343266199

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