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Properties

Label 1-2205-2205.1159-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.965 - 0.260i$
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  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)8-s + (−0.988 + 0.149i)11-s + (0.988 − 0.149i)13-s + (−0.900 − 0.433i)16-s + (0.733 − 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.733 + 0.680i)22-s + (−0.955 + 0.294i)23-s + (−0.733 − 0.680i)26-s + (−0.733 + 0.680i)29-s + 31-s + (0.222 + 0.974i)32-s + (−0.988 − 0.149i)34-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)8-s + (−0.988 + 0.149i)11-s + (0.988 − 0.149i)13-s + (−0.900 − 0.433i)16-s + (0.733 − 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.733 + 0.680i)22-s + (−0.955 + 0.294i)23-s + (−0.733 − 0.680i)26-s + (−0.733 + 0.680i)29-s + 31-s + (0.222 + 0.974i)32-s + (−0.988 − 0.149i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.965 - 0.260i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.965 - 0.260i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06002935662 - 0.4528517681i\)
\(L(\frac12)\) \(\approx\) \(0.06002935662 - 0.4528517681i\)
\(L(1)\) \(\approx\) \(0.6094697463 - 0.2344871480i\)
\(L(1)\) \(\approx\) \(0.6094697463 - 0.2344871480i\)

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 \( 1 + (-0.988 + 0.149i)T \)
13 \( 1 + (0.988 - 0.149i)T \)
17 \( 1 + (0.733 - 0.680i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.955 + 0.294i)T \)
29 \( 1 + (-0.733 + 0.680i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.955 - 0.294i)T \)
41 \( 1 + (0.0747 - 0.997i)T \)
43 \( 1 + (-0.0747 - 0.997i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (-0.955 + 0.294i)T \)
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 \( 1 + (0.988 + 0.149i)T \)
79 \( 1 + T \)
83 \( 1 + (0.988 + 0.149i)T \)
89 \( 1 + (0.365 + 0.930i)T \)
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

−19.78669938296672725552761699552, −19.191641494109277283661109016, −18.483727141446013000053523063387, −17.925869450894612329045068173177, −17.184562911677437729061483986665, −16.402350098363843249056333160944, −15.814656189467619578392758720393, −15.19570747179743182013820396605, −14.446624096892338813495658802023, −13.528553589956018195740986072696, −13.12044165813227870704210119228, −11.93676656358719653618062283984, −10.96626639915765759819969871103, −10.45918603721271518121343185142, −9.671388783426057147600248524597, −8.830672520681474296954295215642, −8.02079045921266779507620654011, −7.71121525004994337360958912773, −6.31622505662804493433715681965, −6.18553191880683951706437493973, −5.071307538264745731626315842604, −4.34367681818435867052186305437, −3.18235724311270930679924418516, −2.05365556334152385220327094520, −1.101722755691207896999996785459, 0.205620231368278051082110012646, 1.46950037830422316788470145651, 2.20601853307987973783603732392, 3.28066597481186975831107329850, 3.798155950433296206224552039375, 4.93457751909550269686200735033, 5.76242280147422087295164119343, 6.8656738667552175217707401642, 7.80483196801873824399078875935, 8.23534939263020252072887427937, 9.11957756000066535183980245587, 9.97402034416642456471383979628, 10.535157632485189128451921197048, 11.17234065483245856505901105624, 12.1934963662735255926409459153, 12.52592868933947590883033382869, 13.60710765912866374930210149020, 13.961564990346454951304676159824, 15.26310974024525563564861655088, 16.003181615684830465852848864889, 16.54552652893622860619813691979, 17.46926349581806707600593285832, 18.12611186507342810867103141182, 18.73584196518184260391918329674, 19.187641287049966074995951427048

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