Properties

Label 2-2205-1.1-c3-0-87
Degree $2$
Conductor $2205$
Sign $1$
Analytic cond. $130.099$
Root an. cond. $11.4061$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.644·2-s − 7.58·4-s + 5·5-s + 10.0·8-s − 3.22·10-s + 47.7·11-s + 57.2·13-s + 54.1·16-s + 36.9·17-s + 30.7·19-s − 37.9·20-s − 30.7·22-s − 53.1·23-s + 25·25-s − 36.8·26-s + 195.·29-s + 257.·31-s − 115.·32-s − 23.8·34-s + 346.·37-s − 19.8·38-s + 50.2·40-s − 267.·41-s − 176.·43-s − 361.·44-s + 34.2·46-s + 311.·47-s + ⋯
L(s)  = 1  − 0.227·2-s − 0.948·4-s + 0.447·5-s + 0.443·8-s − 0.101·10-s + 1.30·11-s + 1.22·13-s + 0.846·16-s + 0.527·17-s + 0.371·19-s − 0.423·20-s − 0.298·22-s − 0.481·23-s + 0.200·25-s − 0.278·26-s + 1.25·29-s + 1.49·31-s − 0.637·32-s − 0.120·34-s + 1.53·37-s − 0.0846·38-s + 0.198·40-s − 1.01·41-s − 0.627·43-s − 1.23·44-s + 0.109·46-s + 0.967·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(130.099\)
Root analytic conductor: \(11.4061\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.370178927\)
\(L(\frac12)\) \(\approx\) \(2.370178927\)
\(L(\frac{5}{2})\) 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 \( 1 - 5T \)
7 \( 1 \)
good2 \( 1 + 0.644T + 8T^{2} \)
11 \( 1 - 47.7T + 1.33e3T^{2} \)
13 \( 1 - 57.2T + 2.19e3T^{2} \)
17 \( 1 - 36.9T + 4.91e3T^{2} \)
19 \( 1 - 30.7T + 6.85e3T^{2} \)
23 \( 1 + 53.1T + 1.21e4T^{2} \)
29 \( 1 - 195.T + 2.43e4T^{2} \)
31 \( 1 - 257.T + 2.97e4T^{2} \)
37 \( 1 - 346.T + 5.06e4T^{2} \)
41 \( 1 + 267.T + 6.89e4T^{2} \)
43 \( 1 + 176.T + 7.95e4T^{2} \)
47 \( 1 - 311.T + 1.03e5T^{2} \)
53 \( 1 - 492.T + 1.48e5T^{2} \)
59 \( 1 + 98.7T + 2.05e5T^{2} \)
61 \( 1 - 82.1T + 2.26e5T^{2} \)
67 \( 1 - 654.T + 3.00e5T^{2} \)
71 \( 1 + 779.T + 3.57e5T^{2} \)
73 \( 1 + 829.T + 3.89e5T^{2} \)
79 \( 1 + 769.T + 4.93e5T^{2} \)
83 \( 1 + 613.T + 5.71e5T^{2} \)
89 \( 1 - 457.T + 7.04e5T^{2} \)
97 \( 1 + 1.41e3T + 9.12e5T^{2} \)
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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

−8.472910149659328707598583390885, −8.424601609982491503104505682829, −7.13441248857988855035043823482, −6.24465937673423538220729028828, −5.65742573053319076506289800745, −4.53495372954292475609337195101, −3.94301767491994167299432621593, −2.95120602018056173510434217557, −1.40934051512823791689215379975, −0.848489568283673992685265416503, 0.848489568283673992685265416503, 1.40934051512823791689215379975, 2.95120602018056173510434217557, 3.94301767491994167299432621593, 4.53495372954292475609337195101, 5.65742573053319076506289800745, 6.24465937673423538220729028828, 7.13441248857988855035043823482, 8.424601609982491503104505682829, 8.472910149659328707598583390885

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