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Properties

Label 1-2205-2205.2003-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.995 - 0.0910i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (−0.294 + 0.955i)2-s + (−0.826 − 0.563i)4-s + (0.781 − 0.623i)8-s + (−0.955 − 0.294i)11-s + (−0.680 − 0.733i)13-s + (0.365 + 0.930i)16-s + (0.433 − 0.900i)17-s − 19-s + (0.563 − 0.826i)22-s + (0.563 − 0.826i)23-s + (0.900 − 0.433i)26-s + (0.826 − 0.563i)29-s + (−0.5 − 0.866i)31-s + (−0.997 + 0.0747i)32-s + (0.733 + 0.680i)34-s + ⋯
L(s)  = 1  + (−0.294 + 0.955i)2-s + (−0.826 − 0.563i)4-s + (0.781 − 0.623i)8-s + (−0.955 − 0.294i)11-s + (−0.680 − 0.733i)13-s + (0.365 + 0.930i)16-s + (0.433 − 0.900i)17-s − 19-s + (0.563 − 0.826i)22-s + (0.563 − 0.826i)23-s + (0.900 − 0.433i)26-s + (0.826 − 0.563i)29-s + (−0.5 − 0.866i)31-s + (−0.997 + 0.0747i)32-s + (0.733 + 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.995 - 0.0910i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.995 - 0.0910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007388708743 + 0.1620472787i\)
\(L(\frac12)\) \(\approx\) \(0.007388708743 + 0.1620472787i\)
\(L(1)\) \(\approx\) \(0.6184601324 + 0.1943526251i\)
\(L(1)\) \(\approx\) \(0.6184601324 + 0.1943526251i\)

Euler product

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

−19.502457841606284047471759391185, −18.74570157854470740708781099847, −18.06411527378007621022819212573, −17.281953946920782698071743680, −16.7789643151775361909448685633, −15.82285103445422590495414240312, −14.87730962315953390509266431131, −14.199612579064822917195320291296, −13.30345426556018752875562383936, −12.66520325060058748856566044833, −12.10700214525726417053364893927, −11.243979150855281919106335752520, −10.3694124280510187947149708224, −10.11000671314065682902618186745, −8.957810002820459504501494299492, −8.50438172183695046985203350155, −7.5178463304848177196675730598, −6.84071904094651289733028170245, −5.47370196585490565775242942079, −4.87025533668863212732623096890, −3.93327837091322929574433563578, −3.15428742713960128811817043914, −2.15405663252318621129481550832, −1.5761979331729037087686831876, −0.06949112592677604444966976134, 0.967015979558222430187185992619, 2.39410365622959505948478149067, 3.23580225379946731021378196594, 4.606488458268450607760971510, 4.97806796105640330513151028515, 5.935311640316538285272820030670, 6.63065805855977854282886839396, 7.58677547786228731995489034246, 8.06687691582157394816103147603, 8.82918865626464840565248134863, 9.80652238293973088535241936501, 10.30052963027013166980862568448, 11.12123060632724653833053513441, 12.26990357878766553454669256410, 13.054075472880812372765886010708, 13.60262852731307435589691310224, 14.55959357713839818128455128085, 15.104899771750967452341118547960, 15.75922269762023426688491647117, 16.57475623470965552317716289472, 17.08558560512323296294097988323, 17.91248644297366065798674262700, 18.58084720825240180250764371516, 19.08262549445596140701334389150, 20.00414900926742296446211853783

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