Properties

Label 2-2205-1.1-c3-0-203
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 $1$

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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2205,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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

−7.73952685814257176208079774274, −7.38180605081587666712307477932, −6.60368509882767943459844027100, −5.65670185350210433770051220262, −4.97944186980841058915908294015, −4.39366655066510195516184319763, −3.50525396743630948966476999391, −2.57163333862389069992405359272, −1.95110997524623837962127854613, 0, 1.95110997524623837962127854613, 2.57163333862389069992405359272, 3.50525396743630948966476999391, 4.39366655066510195516184319763, 4.97944186980841058915908294015, 5.65670185350210433770051220262, 6.60368509882767943459844027100, 7.38180605081587666712307477932, 7.73952685814257176208079774274

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