Properties

Label 2-2205-1.1-c3-0-105
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  − 4.91·2-s + 16.1·4-s − 5·5-s − 39.8·8-s + 24.5·10-s − 12.0·11-s + 46.9·13-s + 66.8·16-s − 44.7·17-s − 31.7·19-s − 80.5·20-s + 58.9·22-s + 39.4·23-s + 25·25-s − 230.·26-s − 87.2·29-s + 304.·31-s − 9.27·32-s + 219.·34-s − 151.·37-s + 155.·38-s + 199.·40-s − 282.·41-s − 143.·43-s − 193.·44-s − 193.·46-s − 88.8·47-s + ⋯
L(s)  = 1  − 1.73·2-s + 2.01·4-s − 0.447·5-s − 1.76·8-s + 0.776·10-s − 0.328·11-s + 1.00·13-s + 1.04·16-s − 0.637·17-s − 0.383·19-s − 0.900·20-s + 0.571·22-s + 0.357·23-s + 0.200·25-s − 1.73·26-s − 0.558·29-s + 1.76·31-s − 0.0512·32-s + 1.10·34-s − 0.671·37-s + 0.665·38-s + 0.787·40-s − 1.07·41-s − 0.507·43-s − 0.662·44-s − 0.620·46-s − 0.275·47-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: \(1\)
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 \( 1 + 4.91T + 8T^{2} \)
11 \( 1 + 12.0T + 1.33e3T^{2} \)
13 \( 1 - 46.9T + 2.19e3T^{2} \)
17 \( 1 + 44.7T + 4.91e3T^{2} \)
19 \( 1 + 31.7T + 6.85e3T^{2} \)
23 \( 1 - 39.4T + 1.21e4T^{2} \)
29 \( 1 + 87.2T + 2.43e4T^{2} \)
31 \( 1 - 304.T + 2.97e4T^{2} \)
37 \( 1 + 151.T + 5.06e4T^{2} \)
41 \( 1 + 282.T + 6.89e4T^{2} \)
43 \( 1 + 143.T + 7.95e4T^{2} \)
47 \( 1 + 88.8T + 1.03e5T^{2} \)
53 \( 1 + 149.T + 1.48e5T^{2} \)
59 \( 1 - 712.T + 2.05e5T^{2} \)
61 \( 1 - 65.0T + 2.26e5T^{2} \)
67 \( 1 - 698.T + 3.00e5T^{2} \)
71 \( 1 - 291.T + 3.57e5T^{2} \)
73 \( 1 - 476.T + 3.89e5T^{2} \)
79 \( 1 - 256.T + 4.93e5T^{2} \)
83 \( 1 + 1.17e3T + 5.71e5T^{2} \)
89 \( 1 + 1.00e3T + 7.04e5T^{2} \)
97 \( 1 + 1.14e3T + 9.12e5T^{2} \)
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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

−8.425571162290477388780433527223, −7.893326562427386680806748335807, −6.84366525560363363172482914472, −6.51213399151245181603647577998, −5.27759087917955294089130096293, −4.09907671594196140534705611449, −2.99886870079497091349977612725, −1.97366932482756549266064292955, −0.973373150579459016674029112394, 0, 0.973373150579459016674029112394, 1.97366932482756549266064292955, 2.99886870079497091349977612725, 4.09907671594196140534705611449, 5.27759087917955294089130096293, 6.51213399151245181603647577998, 6.84366525560363363172482914472, 7.893326562427386680806748335807, 8.425571162290477388780433527223

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