Properties

Label 2-2205-1.1-c3-0-168
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  + 5.51·2-s + 22.4·4-s + 5·5-s + 79.7·8-s + 27.5·10-s − 34.5·11-s + 68.8·13-s + 260.·16-s + 91.4·17-s + 11.8·19-s + 112.·20-s − 190.·22-s + 0.104·23-s + 25·25-s + 380.·26-s − 190.·29-s + 159.·31-s + 798.·32-s + 504.·34-s − 177.·37-s + 65.2·38-s + 398.·40-s − 145.·41-s + 8.25·43-s − 774.·44-s + 0.574·46-s − 260.·47-s + ⋯
L(s)  = 1  + 1.95·2-s + 2.80·4-s + 0.447·5-s + 3.52·8-s + 0.872·10-s − 0.946·11-s + 1.46·13-s + 4.06·16-s + 1.30·17-s + 0.142·19-s + 1.25·20-s − 1.84·22-s + 0.000944·23-s + 0.200·25-s + 2.86·26-s − 1.22·29-s + 0.925·31-s + 4.41·32-s + 2.54·34-s − 0.790·37-s + 0.278·38-s + 1.57·40-s − 0.553·41-s + 0.0292·43-s − 2.65·44-s + 0.00184·46-s − 0.808·47-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

\(L(2)\) \(\approx\) \(10.96933332\)
\(L(\frac12)\) \(\approx\) \(10.96933332\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.51T + 8T^{2} \)
11 \( 1 + 34.5T + 1.33e3T^{2} \)
13 \( 1 - 68.8T + 2.19e3T^{2} \)
17 \( 1 - 91.4T + 4.91e3T^{2} \)
19 \( 1 - 11.8T + 6.85e3T^{2} \)
23 \( 1 - 0.104T + 1.21e4T^{2} \)
29 \( 1 + 190.T + 2.43e4T^{2} \)
31 \( 1 - 159.T + 2.97e4T^{2} \)
37 \( 1 + 177.T + 5.06e4T^{2} \)
41 \( 1 + 145.T + 6.89e4T^{2} \)
43 \( 1 - 8.25T + 7.95e4T^{2} \)
47 \( 1 + 260.T + 1.03e5T^{2} \)
53 \( 1 + 353.T + 1.48e5T^{2} \)
59 \( 1 + 240.T + 2.05e5T^{2} \)
61 \( 1 - 778.T + 2.26e5T^{2} \)
67 \( 1 - 151.T + 3.00e5T^{2} \)
71 \( 1 - 311.T + 3.57e5T^{2} \)
73 \( 1 - 639.T + 3.89e5T^{2} \)
79 \( 1 - 391.T + 4.93e5T^{2} \)
83 \( 1 + 493.T + 5.71e5T^{2} \)
89 \( 1 + 473.T + 7.04e5T^{2} \)
97 \( 1 - 839.T + 9.12e5T^{2} \)
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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

−8.363835170448754283572389860865, −7.68786844051397681431036531713, −6.77893092626830344778664598623, −6.03296467898378840454383364030, −5.46413692874092354570280279476, −4.84351856995033682732403851388, −3.68061146576806708701774357429, −3.23306330096569356934323660889, −2.16717496199725693700868241537, −1.22952304195467700663502412052, 1.22952304195467700663502412052, 2.16717496199725693700868241537, 3.23306330096569356934323660889, 3.68061146576806708701774357429, 4.84351856995033682732403851388, 5.46413692874092354570280279476, 6.03296467898378840454383364030, 6.77893092626830344778664598623, 7.68786844051397681431036531713, 8.363835170448754283572389860865

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