Properties

Label 2-1452-1.1-c3-0-49
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 0.981·5-s + 17.7·7-s + 9·9-s + 61.8·13-s − 2.94·15-s − 130.·17-s − 116.·19-s + 53.3·21-s − 26.4·23-s − 124.·25-s + 27·27-s − 179.·29-s − 319.·31-s − 17.4·35-s − 177.·37-s + 185.·39-s + 338.·41-s − 159.·43-s − 8.83·45-s + 98.9·47-s − 27.0·49-s − 390.·51-s + 562.·53-s − 348.·57-s + 443.·59-s − 551.·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.0877·5-s + 0.959·7-s + 0.333·9-s + 1.32·13-s − 0.0506·15-s − 1.85·17-s − 1.40·19-s + 0.554·21-s − 0.240·23-s − 0.992·25-s + 0.192·27-s − 1.15·29-s − 1.85·31-s − 0.0842·35-s − 0.790·37-s + 0.762·39-s + 1.29·41-s − 0.566·43-s − 0.0292·45-s + 0.307·47-s − 0.0789·49-s − 1.07·51-s + 1.45·53-s − 0.810·57-s + 0.979·59-s − 1.15·61-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: \(1\)
Selberg data: \((2,\ 1452,\ (\ :3/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 - 3T \)
11 \( 1 \)
good5 \( 1 + 0.981T + 125T^{2} \)
7 \( 1 - 17.7T + 343T^{2} \)
13 \( 1 - 61.8T + 2.19e3T^{2} \)
17 \( 1 + 130.T + 4.91e3T^{2} \)
19 \( 1 + 116.T + 6.85e3T^{2} \)
23 \( 1 + 26.4T + 1.21e4T^{2} \)
29 \( 1 + 179.T + 2.43e4T^{2} \)
31 \( 1 + 319.T + 2.97e4T^{2} \)
37 \( 1 + 177.T + 5.06e4T^{2} \)
41 \( 1 - 338.T + 6.89e4T^{2} \)
43 \( 1 + 159.T + 7.95e4T^{2} \)
47 \( 1 - 98.9T + 1.03e5T^{2} \)
53 \( 1 - 562.T + 1.48e5T^{2} \)
59 \( 1 - 443.T + 2.05e5T^{2} \)
61 \( 1 + 551.T + 2.26e5T^{2} \)
67 \( 1 - 454.T + 3.00e5T^{2} \)
71 \( 1 + 214.T + 3.57e5T^{2} \)
73 \( 1 + 162.T + 3.89e5T^{2} \)
79 \( 1 + 167.T + 4.93e5T^{2} \)
83 \( 1 + 357.T + 5.71e5T^{2} \)
89 \( 1 - 1.21e3T + 7.04e5T^{2} \)
97 \( 1 + 1.11e3T + 9.12e5T^{2} \)
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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

−8.765156293362862447454734622859, −8.105843448585047672623932853878, −7.24005520891418966871702063116, −6.34130199880754067088375269365, −5.42026363293499621346039640164, −4.20429019002484175278901395435, −3.82839954627897407391174216343, −2.24894758527071368031501966574, −1.66758701117129657366021418542, 0, 1.66758701117129657366021418542, 2.24894758527071368031501966574, 3.82839954627897407391174216343, 4.20429019002484175278901395435, 5.42026363293499621346039640164, 6.34130199880754067088375269365, 7.24005520891418966871702063116, 8.105843448585047672623932853878, 8.765156293362862447454734622859

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