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Properties

Label 2-2205-21.20-c1-0-19
Degree $2$
Conductor $2205$
Sign $-0.860 - 0.508i$
Analytic cond. $17.6070$
Root an. cond. $4.19607$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.69i·2-s − 0.877·4-s + 5-s + 1.90i·8-s + 1.69i·10-s − 0.421i·11-s + 2.17i·13-s − 4.98·16-s + 5.88·17-s − 1.60i·19-s − 0.877·20-s + 0.715·22-s − 0.156i·23-s + 25-s − 3.68·26-s + ⋯
L(s)  = 1  + 1.19i·2-s − 0.438·4-s + 0.447·5-s + 0.673i·8-s + 0.536i·10-s − 0.127i·11-s + 0.602i·13-s − 1.24·16-s + 1.42·17-s − 0.367i·19-s − 0.196·20-s + 0.152·22-s − 0.0325i·23-s + 0.200·25-s − 0.722·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.860 - 0.508i$
Analytic conductor: \(17.6070\)
Root analytic conductor: \(4.19607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :1/2),\ -0.860 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.985672618\)
\(L(\frac12)\) \(\approx\) \(1.985672618\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good2 \( 1 - 1.69iT - 2T^{2} \)
11 \( 1 + 0.421iT - 11T^{2} \)
13 \( 1 - 2.17iT - 13T^{2} \)
17 \( 1 - 5.88T + 17T^{2} \)
19 \( 1 + 1.60iT - 19T^{2} \)
23 \( 1 + 0.156iT - 23T^{2} \)
29 \( 1 - 6.40iT - 29T^{2} \)
31 \( 1 - 8.33iT - 31T^{2} \)
37 \( 1 + 5.87T + 37T^{2} \)
41 \( 1 - 0.916T + 41T^{2} \)
43 \( 1 - 1.68T + 43T^{2} \)
47 \( 1 - 6.12T + 47T^{2} \)
53 \( 1 + 13.1iT - 53T^{2} \)
59 \( 1 - 8.44T + 59T^{2} \)
61 \( 1 - 6.71iT - 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 16.5iT - 71T^{2} \)
73 \( 1 + 0.804iT - 73T^{2} \)
79 \( 1 - 2.06T + 79T^{2} \)
83 \( 1 - 4.20T + 83T^{2} \)
89 \( 1 - 0.722T + 89T^{2} \)
97 \( 1 - 14.4iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.987206495355046916790081519302, −8.610378336396949097335215444058, −7.62690995280625730588202164956, −6.98209101702201913193718862389, −6.40677954750867078988911792599, −5.38506253320475636016261201769, −5.10411267716795844156742016980, −3.75840409586331702143049383150, −2.65017947497627006077962796871, −1.41168235540183808609424463424, 0.71201666427146431009797039093, 1.81515818485964547541475819061, 2.72768603611412392744028766988, 3.55093839011958010307125687771, 4.42019278967549595104715226577, 5.59708397777136507131549144200, 6.17727221291762810007036501533, 7.35134054752757967053146905033, 7.947936827657056495460371389451, 9.083367344391792076633667619394

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