Properties

Label 2-39e2-1.1-c3-0-148
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  − 3.66·2-s + 5.43·4-s + 9.45·5-s + 15.4·7-s + 9.39·8-s − 34.6·10-s − 12.7·11-s − 56.4·14-s − 77.9·16-s − 18.9·17-s + 95.4·19-s + 51.4·20-s + 46.6·22-s + 104.·23-s − 35.6·25-s + 83.8·28-s + 23.3·29-s − 177.·31-s + 210.·32-s + 69.6·34-s + 145.·35-s − 350.·37-s − 350.·38-s + 88.7·40-s − 348.·41-s + 60.8·43-s − 69.2·44-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.679·4-s + 0.845·5-s + 0.832·7-s + 0.415·8-s − 1.09·10-s − 0.348·11-s − 1.07·14-s − 1.21·16-s − 0.271·17-s + 1.15·19-s + 0.574·20-s + 0.452·22-s + 0.945·23-s − 0.284·25-s + 0.565·28-s + 0.149·29-s − 1.02·31-s + 1.16·32-s + 0.351·34-s + 0.703·35-s − 1.55·37-s − 1.49·38-s + 0.351·40-s − 1.32·41-s + 0.215·43-s − 0.237·44-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 \( 1 + 3.66T + 8T^{2} \)
5 \( 1 - 9.45T + 125T^{2} \)
7 \( 1 - 15.4T + 343T^{2} \)
11 \( 1 + 12.7T + 1.33e3T^{2} \)
17 \( 1 + 18.9T + 4.91e3T^{2} \)
19 \( 1 - 95.4T + 6.85e3T^{2} \)
23 \( 1 - 104.T + 1.21e4T^{2} \)
29 \( 1 - 23.3T + 2.43e4T^{2} \)
31 \( 1 + 177.T + 2.97e4T^{2} \)
37 \( 1 + 350.T + 5.06e4T^{2} \)
41 \( 1 + 348.T + 6.89e4T^{2} \)
43 \( 1 - 60.8T + 7.95e4T^{2} \)
47 \( 1 + 226.T + 1.03e5T^{2} \)
53 \( 1 + 294.T + 1.48e5T^{2} \)
59 \( 1 + 596.T + 2.05e5T^{2} \)
61 \( 1 + 487.T + 2.26e5T^{2} \)
67 \( 1 + 943.T + 3.00e5T^{2} \)
71 \( 1 - 1.04e3T + 3.57e5T^{2} \)
73 \( 1 + 554.T + 3.89e5T^{2} \)
79 \( 1 - 126.T + 4.93e5T^{2} \)
83 \( 1 - 1.44e3T + 5.71e5T^{2} \)
89 \( 1 + 247.T + 7.04e5T^{2} \)
97 \( 1 - 136.T + 9.12e5T^{2} \)
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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

−8.896552912105502014684254008790, −7.975370953055441535160446786446, −7.40434605241084705183376393378, −6.50064403802197746081178154187, −5.32057554153259172178884349756, −4.77646928878874776091293517381, −3.25748067082039095176295830528, −1.93081365299388562573594371294, −1.36391205226987486293852414099, 0, 1.36391205226987486293852414099, 1.93081365299388562573594371294, 3.25748067082039095176295830528, 4.77646928878874776091293517381, 5.32057554153259172178884349756, 6.50064403802197746081178154187, 7.40434605241084705183376393378, 7.975370953055441535160446786446, 8.896552912105502014684254008790

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