Properties

Label 2-39e2-1.1-c3-0-174
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  + 4.22·2-s + 9.85·4-s + 5.85·5-s − 24.1·7-s + 7.85·8-s + 24.7·10-s + 33.8·11-s − 101.·14-s − 45.6·16-s + 49.3·17-s − 76.8·19-s + 57.7·20-s + 143.·22-s − 6.29·23-s − 90.6·25-s − 237.·28-s − 100.·29-s − 307.·31-s − 255.·32-s + 208.·34-s − 141.·35-s − 76.0·37-s − 324.·38-s + 46.0·40-s + 514.·41-s − 268.·43-s + 334.·44-s + ⋯
L(s)  = 1  + 1.49·2-s + 1.23·4-s + 0.524·5-s − 1.30·7-s + 0.347·8-s + 0.783·10-s + 0.928·11-s − 1.94·14-s − 0.713·16-s + 0.704·17-s − 0.927·19-s + 0.645·20-s + 1.38·22-s − 0.0570·23-s − 0.725·25-s − 1.60·28-s − 0.646·29-s − 1.78·31-s − 1.41·32-s + 1.05·34-s − 0.682·35-s − 0.337·37-s − 1.38·38-s + 0.182·40-s + 1.95·41-s − 0.951·43-s + 1.14·44-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 \( 1 - 4.22T + 8T^{2} \)
5 \( 1 - 5.85T + 125T^{2} \)
7 \( 1 + 24.1T + 343T^{2} \)
11 \( 1 - 33.8T + 1.33e3T^{2} \)
17 \( 1 - 49.3T + 4.91e3T^{2} \)
19 \( 1 + 76.8T + 6.85e3T^{2} \)
23 \( 1 + 6.29T + 1.21e4T^{2} \)
29 \( 1 + 100.T + 2.43e4T^{2} \)
31 \( 1 + 307.T + 2.97e4T^{2} \)
37 \( 1 + 76.0T + 5.06e4T^{2} \)
41 \( 1 - 514.T + 6.89e4T^{2} \)
43 \( 1 + 268.T + 7.95e4T^{2} \)
47 \( 1 - 460.T + 1.03e5T^{2} \)
53 \( 1 + 67.8T + 1.48e5T^{2} \)
59 \( 1 + 25.2T + 2.05e5T^{2} \)
61 \( 1 + 588.T + 2.26e5T^{2} \)
67 \( 1 + 1.00e3T + 3.00e5T^{2} \)
71 \( 1 + 895.T + 3.57e5T^{2} \)
73 \( 1 - 968.T + 3.89e5T^{2} \)
79 \( 1 + 119.T + 4.93e5T^{2} \)
83 \( 1 + 480.T + 5.71e5T^{2} \)
89 \( 1 + 1.08e3T + 7.04e5T^{2} \)
97 \( 1 + 16.6T + 9.12e5T^{2} \)
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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

−9.049414423421280876435702247836, −7.51634790585697725852780783280, −6.70572028554428766110033985005, −5.96439711141188480340226964364, −5.61310940247907610774611167648, −4.27954986441463774820334998609, −3.69946158314778940149507540391, −2.85072091404331518070366828806, −1.75649509311200649094213002396, 0, 1.75649509311200649094213002396, 2.85072091404331518070366828806, 3.69946158314778940149507540391, 4.27954986441463774820334998609, 5.61310940247907610774611167648, 5.96439711141188480340226964364, 6.70572028554428766110033985005, 7.51634790585697725852780783280, 9.049414423421280876435702247836

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