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Properties

Label 2-39e2-169.164-c0-0-0
Degree $2$
Conductor $1521$
Sign $0.968 - 0.248i$
Analytic cond. $0.759077$
Root an. cond. $0.871250$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.935 + 0.354i)4-s + (0.307 − 0.509i)7-s + i·13-s + (0.748 + 0.663i)16-s + (0.254 + 0.254i)19-s + (−0.992 − 0.120i)25-s + (0.468 − 0.366i)28-s + (1.12 − 1.43i)31-s + (−0.814 − 0.638i)37-s + (0.475 + 0.0576i)43-s + (0.300 + 0.571i)49-s + (−0.354 + 0.935i)52-s + (−1.81 − 0.447i)61-s + (0.464 + 0.885i)64-s + (−0.646 + 1.43i)67-s + ⋯
L(s)  = 1  + (0.935 + 0.354i)4-s + (0.307 − 0.509i)7-s + i·13-s + (0.748 + 0.663i)16-s + (0.254 + 0.254i)19-s + (−0.992 − 0.120i)25-s + (0.468 − 0.366i)28-s + (1.12 − 1.43i)31-s + (−0.814 − 0.638i)37-s + (0.475 + 0.0576i)43-s + (0.300 + 0.571i)49-s + (−0.354 + 0.935i)52-s + (−1.81 − 0.447i)61-s + (0.464 + 0.885i)64-s + (−0.646 + 1.43i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.418150404\)
\(L(\frac12)\) \(\approx\) \(1.418150404\)
\(L(1)\) 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 \( 1 - iT \)
good2 \( 1 + (-0.935 - 0.354i)T^{2} \)
5 \( 1 + (0.992 + 0.120i)T^{2} \)
7 \( 1 + (-0.307 + 0.509i)T + (-0.464 - 0.885i)T^{2} \)
11 \( 1 + (-0.935 + 0.354i)T^{2} \)
17 \( 1 + (-0.885 + 0.464i)T^{2} \)
19 \( 1 + (-0.254 - 0.254i)T + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.354 + 0.935i)T^{2} \)
31 \( 1 + (-1.12 + 1.43i)T + (-0.239 - 0.970i)T^{2} \)
37 \( 1 + (0.814 + 0.638i)T + (0.239 + 0.970i)T^{2} \)
41 \( 1 + (-0.822 + 0.568i)T^{2} \)
43 \( 1 + (-0.475 - 0.0576i)T + (0.970 + 0.239i)T^{2} \)
47 \( 1 + (0.663 + 0.748i)T^{2} \)
53 \( 1 + (0.885 - 0.464i)T^{2} \)
59 \( 1 + (-0.992 - 0.120i)T^{2} \)
61 \( 1 + (1.81 + 0.447i)T + (0.885 + 0.464i)T^{2} \)
67 \( 1 + (0.646 - 1.43i)T + (-0.663 - 0.748i)T^{2} \)
71 \( 1 + (-0.822 + 0.568i)T^{2} \)
73 \( 1 + (0.359 + 1.96i)T + (-0.935 + 0.354i)T^{2} \)
79 \( 1 + (-0.329 - 0.869i)T + (-0.748 + 0.663i)T^{2} \)
83 \( 1 + (0.822 + 0.568i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (0.0853 + 1.41i)T + (-0.992 + 0.120i)T^{2} \)
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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

−9.763900564170151476414334835629, −8.859424930997272222051750353895, −7.82517381220791152752246033130, −7.43089818897200363003043670508, −6.47243084649943103669501872976, −5.82023762333231681541958199347, −4.49264402368627410999809531599, −3.75271899941653833902803484552, −2.55820543612008146758487262612, −1.56831373771656897959379478292, 1.39280525603450894595208391865, 2.55413599380054132909581364369, 3.37633506164023262929340337286, 4.84203828121974990094480624226, 5.58894766906603301932229128298, 6.32600022573442398231441729452, 7.22520887836413763983405354472, 7.974706852097250917348501156279, 8.752741630650816711685785082870, 9.797312403934016578589150588753

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