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Properties

Label 2-39e2-13.7-c0-0-1
Degree $2$
Conductor $1521$
Sign $0.674 - 0.738i$
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  + (−0.866 + 0.5i)4-s + (1.36 + 0.366i)7-s + (0.499 − 0.866i)16-s + (−0.366 + 1.36i)19-s − i·25-s + (−1.36 + 0.366i)28-s + (1 + i)31-s + (0.366 + 1.36i)37-s + (0.866 + 0.5i)49-s + 0.999i·64-s + (1.36 − 0.366i)67-s + (−1 + i)73-s + (−0.366 − 1.36i)76-s + (0.366 − 1.36i)97-s + (0.5 + 0.866i)100-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)4-s + (1.36 + 0.366i)7-s + (0.499 − 0.866i)16-s + (−0.366 + 1.36i)19-s − i·25-s + (−1.36 + 0.366i)28-s + (1 + i)31-s + (0.366 + 1.36i)37-s + (0.866 + 0.5i)49-s + 0.999i·64-s + (1.36 − 0.366i)67-s + (−1 + i)73-s + (−0.366 − 1.36i)76-s + (0.366 − 1.36i)97-s + (0.5 + 0.866i)100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.674 - 0.738i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.674 - 0.738i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034586889\)
\(L(\frac12)\) \(\approx\) \(1.034586889\)
\(L(1)\) 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 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T^{2} \)
5 \( 1 + iT^{2} \)
7 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1 - i)T + iT^{2} \)
37 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.789983730860939749090604346396, −8.635401092113366327746898034725, −8.309615159805725908289861330521, −7.71047864039964680507588133831, −6.50127339193516017243931254201, −5.44727529413500105361017666323, −4.71105790625125623305125133738, −4.01381583352518620004423396348, −2.80054720559142132140513939114, −1.46383854567660906090243976228, 1.00590992146060072376580203962, 2.27257178374952788949614248738, 3.83953811371073736117525611212, 4.64762352851899894365460135792, 5.19795534094704571975896521050, 6.17477414005074460699207809921, 7.30603228816637395553731706194, 8.027026064507285190245537838033, 8.827211079583198201941644455174, 9.431850757414943487974547202953

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