Properties

Label 2-2205-1.1-c3-0-127
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  + 5.02·2-s + 17.2·4-s + 5·5-s + 46.5·8-s + 25.1·10-s + 0.888·11-s − 25.9·13-s + 95.6·16-s + 96.3·17-s − 89.0·19-s + 86.2·20-s + 4.46·22-s + 116.·23-s + 25·25-s − 130.·26-s + 222.·29-s − 12.9·31-s + 108.·32-s + 483.·34-s + 91.2·37-s − 447.·38-s + 232.·40-s + 98.4·41-s + 392.·43-s + 15.3·44-s + 585.·46-s + 220.·47-s + ⋯
L(s)  = 1  + 1.77·2-s + 2.15·4-s + 0.447·5-s + 2.05·8-s + 0.794·10-s + 0.0243·11-s − 0.553·13-s + 1.49·16-s + 1.37·17-s − 1.07·19-s + 0.964·20-s + 0.0432·22-s + 1.05·23-s + 0.200·25-s − 0.983·26-s + 1.42·29-s − 0.0748·31-s + 0.600·32-s + 2.44·34-s + 0.405·37-s − 1.91·38-s + 0.919·40-s + 0.374·41-s + 1.39·43-s + 0.0525·44-s + 1.87·46-s + 0.683·47-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: \(0\)
Selberg data: \((2,\ 2205,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(8.552756770\)
\(L(\frac12)\) \(\approx\) \(8.552756770\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.02T + 8T^{2} \)
11 \( 1 - 0.888T + 1.33e3T^{2} \)
13 \( 1 + 25.9T + 2.19e3T^{2} \)
17 \( 1 - 96.3T + 4.91e3T^{2} \)
19 \( 1 + 89.0T + 6.85e3T^{2} \)
23 \( 1 - 116.T + 1.21e4T^{2} \)
29 \( 1 - 222.T + 2.43e4T^{2} \)
31 \( 1 + 12.9T + 2.97e4T^{2} \)
37 \( 1 - 91.2T + 5.06e4T^{2} \)
41 \( 1 - 98.4T + 6.89e4T^{2} \)
43 \( 1 - 392.T + 7.95e4T^{2} \)
47 \( 1 - 220.T + 1.03e5T^{2} \)
53 \( 1 - 228.T + 1.48e5T^{2} \)
59 \( 1 + 13.5T + 2.05e5T^{2} \)
61 \( 1 - 205.T + 2.26e5T^{2} \)
67 \( 1 - 325.T + 3.00e5T^{2} \)
71 \( 1 - 583.T + 3.57e5T^{2} \)
73 \( 1 + 950.T + 3.89e5T^{2} \)
79 \( 1 - 451.T + 4.93e5T^{2} \)
83 \( 1 + 164.T + 5.71e5T^{2} \)
89 \( 1 + 884.T + 7.04e5T^{2} \)
97 \( 1 - 62.1T + 9.12e5T^{2} \)
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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

−8.601757916538448742638715500492, −7.56330323277182013915655238235, −6.86102769200083666831633887865, −6.09018890353227476352897142689, −5.44304587000889713746456498017, −4.71444468899495033503642268457, −3.95098051026681906349955230434, −2.93471060463403888453952998932, −2.33448684078510403649306676377, −1.05066349423197204254979750577, 1.05066349423197204254979750577, 2.33448684078510403649306676377, 2.93471060463403888453952998932, 3.95098051026681906349955230434, 4.71444468899495033503642268457, 5.44304587000889713746456498017, 6.09018890353227476352897142689, 6.86102769200083666831633887865, 7.56330323277182013915655238235, 8.601757916538448742638715500492

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