Properties

Label 2-39e2-13.12-c1-0-18
Degree $2$
Conductor $1521$
Sign $-0.277 - 0.960i$
Analytic cond. $12.1452$
Root an. cond. $3.48500$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 3.46i·5-s + 1.73i·7-s + 3.46i·11-s + 4·16-s − 3.46i·19-s + 6.92i·20-s + 6·23-s − 6.99·25-s + 3.46i·28-s − 6·29-s − 1.73i·31-s − 5.99·35-s + 6.92i·41-s − 43-s + 6.92i·44-s + ⋯
L(s)  = 1  + 4-s + 1.54i·5-s + 0.654i·7-s + 1.04i·11-s + 16-s − 0.794i·19-s + 1.54i·20-s + 1.25·23-s − 1.39·25-s + 0.654i·28-s − 1.11·29-s − 0.311i·31-s − 1.01·35-s + 1.08i·41-s − 0.152·43-s + 1.04i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(12.1452\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ -0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.079320168\)
\(L(\frac12)\) \(\approx\) \(2.079320168\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T^{2} \)
5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + 3.46iT - 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 - 8.66iT - 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−9.840291780894637970062636974721, −9.035578530809179338641130677321, −7.72262467738625320139130400839, −7.22821951012821342910087383198, −6.58144726406100902676289329665, −5.88536566003845186685257809154, −4.74812219646033705570892151709, −3.32399821595725790197619322353, −2.69014699240501934300604652367, −1.85160858216345864639681883169, 0.802733956456812466215252951307, 1.72960496393829983522528904597, 3.19996939244799445412646580023, 4.09028629176335646176633137232, 5.25056178523445820072042224915, 5.80923734757000170175100448326, 6.86764579918031861030213549245, 7.68413198227570546914699010292, 8.430514596693369351002158169863, 9.102409662652452915767317586732

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