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Properties

Label 1-2205-2205.1237-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.800 - 0.599i$
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  + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (0.433 + 0.900i)8-s + (0.365 − 0.930i)11-s + (0.930 + 0.365i)13-s + (−0.900 − 0.433i)16-s + (0.294 − 0.955i)17-s + (−0.5 − 0.866i)19-s + (0.294 + 0.955i)22-s + (−0.680 + 0.733i)23-s + (−0.955 + 0.294i)26-s + (−0.955 − 0.294i)29-s − 31-s + (0.974 − 0.222i)32-s + (0.365 + 0.930i)34-s + ⋯
L(s)  = 1  + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (0.433 + 0.900i)8-s + (0.365 − 0.930i)11-s + (0.930 + 0.365i)13-s + (−0.900 − 0.433i)16-s + (0.294 − 0.955i)17-s + (−0.5 − 0.866i)19-s + (0.294 + 0.955i)22-s + (−0.680 + 0.733i)23-s + (−0.955 + 0.294i)26-s + (−0.955 − 0.294i)29-s − 31-s + (0.974 − 0.222i)32-s + (0.365 + 0.930i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.800 - 0.599i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.800 - 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08571560552 - 0.2575241797i\)
\(L(\frac12)\) \(\approx\) \(0.08571560552 - 0.2575241797i\)
\(L(1)\) \(\approx\) \(0.6306475990 + 0.03899356191i\)
\(L(1)\) \(\approx\) \(0.6306475990 + 0.03899356191i\)

Euler product

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

−20.140187804842711758054482669984, −19.26026289888280961391177856721, −18.48450904456935824786180954434, −18.10480980646094693616873569533, −17.10055408614232989343309550782, −16.74087350097521642606173612373, −15.830814446956646443716993338563, −15.02949537766636567531384320045, −14.26748389628688739839813561774, −13.11736716280212095006547447264, −12.642744180655652481750006278213, −11.974691688919935089365169909021, −11.07040338106830514021700543241, −10.4178255969691111914429880507, −9.83363521393059300200779994940, −8.94088416171733968585649003874, −8.24008009815599548909878758046, −7.626628149177364266759593717727, −6.61871631027718135773659788538, −5.942362760463403376103660871491, −4.61348755318203268254805844374, −3.80551129671866057922675824438, −3.16433871200126563956776390692, −1.781551108769977449564008580930, −1.55941743415618869618051560633, 0.11814361833786726460284284775, 1.2559343677097385260357163065, 2.14321189633244990036359112108, 3.35032044950368383431172141829, 4.26237458189329105798730117867, 5.50993832548790612534263098640, 5.83807113418116605894656633346, 6.93863689195338252409967085549, 7.3744473433363188246478801382, 8.549857565491150420825207949128, 8.87236065443483563259621700853, 9.66104533780465759037571032934, 10.59410054773428801228685541005, 11.28469383039160013049547481767, 11.792246948058251840339903979909, 13.15207365618013157857895548855, 13.85621614892087567079956726722, 14.33987707483343148365547526363, 15.44625611370439393348808156761, 15.84028334214951323514105442144, 16.64600859783757964849024126405, 17.17045758418818612415756791884, 18.06474860292303764881502589358, 18.69432815716931736511142899160, 19.16610018778367248435902205425

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