Properties

Label 2-2205-1.1-c1-0-25
Degree $2$
Conductor $2205$
Sign $1$
Analytic cond. $17.6070$
Root an. cond. $4.19607$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 5-s − 3·8-s + 10-s + 4·11-s + 2·13-s − 16-s + 2·17-s − 4·19-s − 20-s + 4·22-s + 25-s + 2·26-s + 2·29-s + 5·32-s + 2·34-s − 10·37-s − 4·38-s − 3·40-s + 10·41-s + 4·43-s − 4·44-s + 8·47-s + 50-s − 2·52-s + 10·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s + 0.316·10-s + 1.20·11-s + 0.554·13-s − 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.392·26-s + 0.371·29-s + 0.883·32-s + 0.342·34-s − 1.64·37-s − 0.648·38-s − 0.474·40-s + 1.56·41-s + 0.609·43-s − 0.603·44-s + 1.16·47-s + 0.141·50-s − 0.277·52-s + 1.37·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(17.6070\)
Root analytic conductor: \(4.19607\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2205} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.445093127\)
\(L(\frac12)\) \(\approx\) \(2.445093127\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.890589428908360265391160029852, −8.627000477337622059216224790867, −7.37485778338999061621083536758, −6.39356342307905775022074491331, −5.91397914498304472267826736485, −5.03119182641673168471146566272, −4.09922216243454775532088496022, −3.56571005686312005410318267112, −2.32940333880865099555943992805, −0.964820104627525128067217267144, 0.964820104627525128067217267144, 2.32940333880865099555943992805, 3.56571005686312005410318267112, 4.09922216243454775532088496022, 5.03119182641673168471146566272, 5.91397914498304472267826736485, 6.39356342307905775022074491331, 7.37485778338999061621083536758, 8.627000477337622059216224790867, 8.890589428908360265391160029852

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