Properties

Label 2-1452-1.1-c3-0-16
Degree $2$
Conductor $1452$
Sign $1$
Analytic cond. $85.6707$
Root an. cond. $9.25585$
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  + 3·3-s − 10.3·5-s + 18.4·7-s + 9·9-s + 93.4·13-s − 30.9·15-s − 75.0·17-s + 38.2·19-s + 55.2·21-s − 107.·23-s − 18.8·25-s + 27·27-s + 185.·29-s + 285.·31-s − 189.·35-s + 35.8·37-s + 280.·39-s − 446.·41-s + 295.·43-s − 92.7·45-s − 66.3·47-s − 4.12·49-s − 225.·51-s + 430.·53-s + 114.·57-s − 382.·59-s + 277.·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.921·5-s + 0.993·7-s + 0.333·9-s + 1.99·13-s − 0.532·15-s − 1.07·17-s + 0.461·19-s + 0.573·21-s − 0.973·23-s − 0.150·25-s + 0.192·27-s + 1.18·29-s + 1.65·31-s − 0.916·35-s + 0.159·37-s + 1.15·39-s − 1.69·41-s + 1.04·43-s − 0.307·45-s − 0.206·47-s − 0.0120·49-s − 0.618·51-s + 1.11·53-s + 0.266·57-s − 0.844·59-s + 0.582·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(85.6707\)
Root analytic conductor: \(9.25585\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.834354134\)
\(L(\frac12)\) \(\approx\) \(2.834354134\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 3T \)
11 \( 1 \)
good5 \( 1 + 10.3T + 125T^{2} \)
7 \( 1 - 18.4T + 343T^{2} \)
13 \( 1 - 93.4T + 2.19e3T^{2} \)
17 \( 1 + 75.0T + 4.91e3T^{2} \)
19 \( 1 - 38.2T + 6.85e3T^{2} \)
23 \( 1 + 107.T + 1.21e4T^{2} \)
29 \( 1 - 185.T + 2.43e4T^{2} \)
31 \( 1 - 285.T + 2.97e4T^{2} \)
37 \( 1 - 35.8T + 5.06e4T^{2} \)
41 \( 1 + 446.T + 6.89e4T^{2} \)
43 \( 1 - 295.T + 7.95e4T^{2} \)
47 \( 1 + 66.3T + 1.03e5T^{2} \)
53 \( 1 - 430.T + 1.48e5T^{2} \)
59 \( 1 + 382.T + 2.05e5T^{2} \)
61 \( 1 - 277.T + 2.26e5T^{2} \)
67 \( 1 + 363.T + 3.00e5T^{2} \)
71 \( 1 - 991.T + 3.57e5T^{2} \)
73 \( 1 + 223.T + 3.89e5T^{2} \)
79 \( 1 + 310.T + 4.93e5T^{2} \)
83 \( 1 + 1.07e3T + 5.71e5T^{2} \)
89 \( 1 + 519.T + 7.04e5T^{2} \)
97 \( 1 - 1.09e3T + 9.12e5T^{2} \)
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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

−8.719618791303312960077674403684, −8.401274162404501390068298805364, −7.84382692680662178305256979658, −6.79684170948227500311499916023, −5.95702686102702255678771948414, −4.64469852346036868989481359335, −4.07887964698170825706428560282, −3.17482008660312470183075688649, −1.89796751765664286378818172580, −0.843433896825307915882105291124, 0.843433896825307915882105291124, 1.89796751765664286378818172580, 3.17482008660312470183075688649, 4.07887964698170825706428560282, 4.64469852346036868989481359335, 5.95702686102702255678771948414, 6.79684170948227500311499916023, 7.84382692680662178305256979658, 8.401274162404501390068298805364, 8.719618791303312960077674403684

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