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Properties

Label 1-2205-2205.1964-r1-0-0
Degree $1$
Conductor $2205$
Sign $0.996 - 0.0889i$
Analytic cond. $236.960$
Root an. cond. $236.960$
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  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)8-s + (−0.365 − 0.930i)11-s + (−0.365 − 0.930i)13-s + (−0.900 + 0.433i)16-s + (0.955 − 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.955 − 0.294i)22-s + (−0.733 + 0.680i)23-s + (−0.955 − 0.294i)26-s + (−0.955 + 0.294i)29-s + 31-s + (−0.222 + 0.974i)32-s + (0.365 − 0.930i)34-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)8-s + (−0.365 − 0.930i)11-s + (−0.365 − 0.930i)13-s + (−0.900 + 0.433i)16-s + (0.955 − 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.955 − 0.294i)22-s + (−0.733 + 0.680i)23-s + (−0.955 − 0.294i)26-s + (−0.955 + 0.294i)29-s + 31-s + (−0.222 + 0.974i)32-s + (0.365 − 0.930i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.996 - 0.0889i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1964, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ 0.996 - 0.0889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.446030079 - 0.06442519422i\)
\(L(\frac12)\) \(\approx\) \(1.446030079 - 0.06442519422i\)
\(L(1)\) \(\approx\) \(1.020367120 - 0.6003059403i\)
\(L(1)\) \(\approx\) \(1.020367120 - 0.6003059403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (-0.365 - 0.930i)T \)
13 \( 1 + (-0.365 - 0.930i)T \)
17 \( 1 + (0.955 - 0.294i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 + T \)
37 \( 1 + (0.733 + 0.680i)T \)
41 \( 1 + (-0.826 + 0.563i)T \)
43 \( 1 + (-0.826 - 0.563i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (-0.733 + 0.680i)T \)
59 \( 1 + (0.900 - 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + T \)
83 \( 1 + (0.365 - 0.930i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−19.54032033403541934071888982991, −18.73292927101521576597018773669, −17.927313500637551172878413794996, −17.25196396614588007371189354881, −16.60719718552338791781453807873, −15.945265164541035339688957631027, −15.03403100820279742975519718232, −14.68713011928757625744557923124, −13.817320369734640508374914015238, −13.12339359685141895777627427232, −12.32327132014643824977866389646, −11.864147054170028763613174253901, −10.83041538758882098195205878647, −9.79810904603013353968600673377, −9.16086424427063849752340240301, −8.147509896901610888728967238624, −7.57010145328201440243806804648, −6.73839111114326137008712746612, −6.12909164795744179013144164650, −5.08838475113845534536614470991, −4.51267120268840970454255793715, −3.75546010889986847068529030689, −2.65040861565312818094182775448, −1.890208523245150766705882717571, −0.23967896247117087963556875527, 0.75699989030022973750231389897, 1.66292715798767812494788071590, 2.76346886847241626057500850287, 3.35276418008303817131531282909, 4.15184155565577005445426646605, 5.31736724752703901149741913256, 5.63720822171755349613935670693, 6.54412930286166093762564014114, 7.77155476215894105194325973492, 8.3763373781336230089438331453, 9.48626820648203400502445571037, 10.199397159766661432700573584215, 10.656402402839679443612720705190, 11.75071534547008001013077102458, 12.05904600127427878573776496008, 13.13626746284038779298403510000, 13.49228827365029900481493259555, 14.42740020223537067140698640559, 14.98643685653198297455618499643, 15.79663936317366351479416495699, 16.61119900030592823095109100447, 17.46395770326569712455889089395, 18.56749689788451599658873895489, 18.7245641189667916243128011930, 19.66955932221427718469597481578

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