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Properties

Label 1-2205-2205.43-r1-0-0
Degree $1$
Conductor $2205$
Sign $0.709 - 0.704i$
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  + (0.680 + 0.733i)2-s + (−0.0747 + 0.997i)4-s + (−0.781 + 0.623i)8-s + (−0.733 + 0.680i)11-s + (−0.294 + 0.955i)13-s + (−0.988 − 0.149i)16-s + (−0.433 + 0.900i)17-s − 19-s + (−0.997 − 0.0747i)22-s + (0.997 + 0.0747i)23-s + (−0.900 + 0.433i)26-s + (−0.0747 − 0.997i)29-s + (−0.5 + 0.866i)31-s + (−0.563 − 0.826i)32-s + (−0.955 + 0.294i)34-s + ⋯
L(s)  = 1  + (0.680 + 0.733i)2-s + (−0.0747 + 0.997i)4-s + (−0.781 + 0.623i)8-s + (−0.733 + 0.680i)11-s + (−0.294 + 0.955i)13-s + (−0.988 − 0.149i)16-s + (−0.433 + 0.900i)17-s − 19-s + (−0.997 − 0.0747i)22-s + (0.997 + 0.0747i)23-s + (−0.900 + 0.433i)26-s + (−0.0747 − 0.997i)29-s + (−0.5 + 0.866i)31-s + (−0.563 − 0.826i)32-s + (−0.955 + 0.294i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.709 - 0.704i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ 0.709 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1637514960 + 0.06745788848i\)
\(L(\frac12)\) \(\approx\) \(-0.1637514960 + 0.06745788848i\)
\(L(1)\) \(\approx\) \(0.8467828150 + 0.7124200673i\)
\(L(1)\) \(\approx\) \(0.8467828150 + 0.7124200673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.680 + 0.733i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (-0.294 + 0.955i)T \)
17 \( 1 + (-0.433 + 0.900i)T \)
19 \( 1 - T \)
23 \( 1 + (0.997 + 0.0747i)T \)
29 \( 1 + (-0.0747 - 0.997i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.433 + 0.900i)T \)
41 \( 1 + (-0.988 + 0.149i)T \)
43 \( 1 + (0.149 - 0.988i)T \)
47 \( 1 + (0.680 + 0.733i)T \)
53 \( 1 + (-0.433 - 0.900i)T \)
59 \( 1 + (-0.365 - 0.930i)T \)
61 \( 1 + (0.0747 + 0.997i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (-0.974 - 0.222i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.294 + 0.955i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−18.956678419021892590149363819861, −18.45114441543719414603271652474, −17.67185325602031696962424744077, −16.694637569159468679645277794681, −15.815278508656542639140343875467, −15.17180841295921299963965469053, −14.5330364607514021301772162026, −13.61694965380328631342078047640, −13.05920602284969804377608814879, −12.50450416344525430044202206598, −11.57458869868967049657942451203, −10.73903424125220782091471821288, −10.510406514354198267727082634, −9.35500165931460514696813443109, −8.74668470725611928834269264547, −7.66712117044032355913112925986, −6.78557586486450652905423707050, −5.82022928962872820057946554419, −5.19617071160555173566786385761, −4.48493482984205201055543200730, −3.3804320592729306457681100531, −2.812230751011182995046191941643, −1.962943460076108814541467795800, −0.73035017007199291899097463233, −0.02848045490257285844037678748, 1.77275984895034664876806707507, 2.52499402922951536241116869069, 3.59408191600651191751593813629, 4.437305917594684257159319172627, 4.97897601505395837959744538085, 5.94095700094253602535849266855, 6.77916563037402752764020387657, 7.24830300796845928344119376171, 8.30024374615235434480358185561, 8.7990801058760931434494558210, 9.82425313017166013878547185781, 10.74593714122304504946748445019, 11.574922646805776441576030030896, 12.44123041293856455535977295333, 12.95489093317465265590992625463, 13.67239775865337502313684750108, 14.499799764214141208637000892172, 15.19606974991048419018437547664, 15.60043409341622483385172488622, 16.613347114743171998997409407390, 17.18837115648099026375314359682, 17.7125085955338263347830616669, 18.7675875045823918073111622347, 19.33779856953103133483762359068, 20.49568931353781461342666183436

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