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Properties

Label 1-2205-2205.1073-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.624 - 0.780i$
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  + (0.680 − 0.733i)2-s + (−0.0747 − 0.997i)4-s + (−0.781 − 0.623i)8-s + (0.222 − 0.974i)11-s + (0.680 − 0.733i)13-s + (−0.988 + 0.149i)16-s + (0.997 + 0.0747i)17-s + (0.5 + 0.866i)19-s + (−0.563 − 0.826i)22-s + (−0.433 + 0.900i)23-s + (−0.0747 − 0.997i)26-s + (0.826 + 0.563i)29-s + (−0.5 − 0.866i)31-s + (−0.563 + 0.826i)32-s + (0.733 − 0.680i)34-s + ⋯
L(s)  = 1  + (0.680 − 0.733i)2-s + (−0.0747 − 0.997i)4-s + (−0.781 − 0.623i)8-s + (0.222 − 0.974i)11-s + (0.680 − 0.733i)13-s + (−0.988 + 0.149i)16-s + (0.997 + 0.0747i)17-s + (0.5 + 0.866i)19-s + (−0.563 − 0.826i)22-s + (−0.433 + 0.900i)23-s + (−0.0747 − 0.997i)26-s + (0.826 + 0.563i)29-s + (−0.5 − 0.866i)31-s + (−0.563 + 0.826i)32-s + (0.733 − 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.624 - 0.780i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1073, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.624 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066180559 - 2.218145347i\)
\(L(\frac12)\) \(\approx\) \(1.066180559 - 2.218145347i\)
\(L(1)\) \(\approx\) \(1.269741872 - 0.9255923544i\)
\(L(1)\) \(\approx\) \(1.269741872 - 0.9255923544i\)

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 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (0.680 - 0.733i)T \)
17 \( 1 + (0.997 + 0.0747i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.433 + 0.900i)T \)
29 \( 1 + (0.826 + 0.563i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.563 - 0.826i)T \)
41 \( 1 + (-0.365 - 0.930i)T \)
43 \( 1 + (0.930 + 0.365i)T \)
47 \( 1 + (0.680 - 0.733i)T \)
53 \( 1 + (-0.563 - 0.826i)T \)
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 \( 1 + (-0.294 + 0.955i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.680 + 0.733i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
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

−20.21178762706359091477633386338, −19.27131261776096344557954905711, −18.235567172282759336381159796116, −17.842770217982928930627650995120, −16.863235343582865628812586289820, −16.36980467808306137418490592030, −15.58772515945217235693244255737, −14.954003194193170803199218616623, −14.12283297245409135569524382876, −13.71728071881639251529065029100, −12.716550308397929676454446571996, −12.1313658252587397000609256748, −11.50082423312679069652883901167, −10.45703889017154688254184705248, −9.47568117396856713802329766327, −8.79794395600047359966592025655, −7.90295408570909279504762356755, −7.19110495229247084764218940573, −6.47966790677863355481820610136, −5.77852702185332500772828685056, −4.69945542093320614000070058597, −4.32270825423494596693467382058, −3.25078920410004550630842116814, −2.45092985345312462725472405250, −1.2221398975442323081445395205, 0.72267810328713408726627487641, 1.4977021061145728531413891477, 2.61581466826623511084074648551, 3.64389999353898358385705703469, 3.76142830464374706254011533588, 5.28249913329171473790055544693, 5.65811257673072028192818828776, 6.39152008762613712937997457484, 7.59954745064213442914828135198, 8.38035320619171130680159190069, 9.346225584037888336421961488321, 10.05196926611699804629689481201, 10.82114228999668217772925900931, 11.423424307320278240149013215277, 12.23035064240334204733771115167, 12.839114393534936611793168332906, 13.744829766289022257538308673866, 14.15753070873119555972186858606, 14.93874554931927136320629762268, 15.885052757095425145969076639262, 16.32470557964742385324939907378, 17.46777508139182598319995369580, 18.28845668373274048113160298092, 18.89788626423939998519447668147, 19.50746547949028908893629080602

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