Properties

Label 2-39e2-1.1-c3-0-80
Degree $2$
Conductor $1521$
Sign $-1$
Analytic cond. $89.7419$
Root an. cond. $9.47322$
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  − 3.73·2-s + 5.95·4-s − 3.90·5-s − 36.4·7-s + 7.64·8-s + 14.5·10-s + 19.1·11-s + 136.·14-s − 76.1·16-s + 83.8·17-s − 46.8·19-s − 23.2·20-s − 71.6·22-s − 103.·23-s − 109.·25-s − 216.·28-s − 108.·29-s + 147.·31-s + 223.·32-s − 313.·34-s + 142.·35-s + 160.·37-s + 175.·38-s − 29.8·40-s + 231.·41-s − 340.·43-s + 114.·44-s + ⋯
L(s)  = 1  − 1.32·2-s + 0.744·4-s − 0.349·5-s − 1.96·7-s + 0.337·8-s + 0.461·10-s + 0.526·11-s + 2.59·14-s − 1.19·16-s + 1.19·17-s − 0.565·19-s − 0.260·20-s − 0.694·22-s − 0.941·23-s − 0.877·25-s − 1.46·28-s − 0.693·29-s + 0.854·31-s + 1.23·32-s − 1.58·34-s + 0.687·35-s + 0.710·37-s + 0.747·38-s − 0.118·40-s + 0.881·41-s − 1.20·43-s + 0.391·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(89.7419\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ -1)\)

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 \( 1 \)
13 \( 1 \)
good2 \( 1 + 3.73T + 8T^{2} \)
5 \( 1 + 3.90T + 125T^{2} \)
7 \( 1 + 36.4T + 343T^{2} \)
11 \( 1 - 19.1T + 1.33e3T^{2} \)
17 \( 1 - 83.8T + 4.91e3T^{2} \)
19 \( 1 + 46.8T + 6.85e3T^{2} \)
23 \( 1 + 103.T + 1.21e4T^{2} \)
29 \( 1 + 108.T + 2.43e4T^{2} \)
31 \( 1 - 147.T + 2.97e4T^{2} \)
37 \( 1 - 160.T + 5.06e4T^{2} \)
41 \( 1 - 231.T + 6.89e4T^{2} \)
43 \( 1 + 340.T + 7.95e4T^{2} \)
47 \( 1 - 119.T + 1.03e5T^{2} \)
53 \( 1 - 732.T + 1.48e5T^{2} \)
59 \( 1 + 229.T + 2.05e5T^{2} \)
61 \( 1 - 108.T + 2.26e5T^{2} \)
67 \( 1 + 10.3T + 3.00e5T^{2} \)
71 \( 1 + 869.T + 3.57e5T^{2} \)
73 \( 1 - 1.09e3T + 3.89e5T^{2} \)
79 \( 1 - 140.T + 4.93e5T^{2} \)
83 \( 1 + 159.T + 5.71e5T^{2} \)
89 \( 1 - 1.06e3T + 7.04e5T^{2} \)
97 \( 1 + 858.T + 9.12e5T^{2} \)
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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

−8.852408024740362329321133654473, −7.987193047777157911475014431801, −7.28602911265057059055142842483, −6.46240351099753395076800363429, −5.78042120064467830866333698018, −4.18927408566201933598434553611, −3.46933826248398265101768088614, −2.28156326127550384971913573599, −0.873477208503307636792463945465, 0, 0.873477208503307636792463945465, 2.28156326127550384971913573599, 3.46933826248398265101768088614, 4.18927408566201933598434553611, 5.78042120064467830866333698018, 6.46240351099753395076800363429, 7.28602911265057059055142842483, 7.987193047777157911475014431801, 8.852408024740362329321133654473

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