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Properties

Label 2-39e2-13.11-c0-0-2
Degree $2$
Conductor $1521$
Sign $0.970 + 0.240i$
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 + (−0.366 − 1.36i)7-s + (0.499 + 0.866i)16-s + (1.36 − 0.366i)19-s − i·25-s + (0.366 − 1.36i)28-s + (1 + i)31-s + (−1.36 − 0.366i)37-s + (−0.866 + 0.5i)49-s + 0.999i·64-s + (−0.366 + 1.36i)67-s + (−1 + i)73-s + (1.36 + 0.366i)76-s + (−1.36 + 0.366i)97-s + (0.5 − 0.866i)100-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)4-s + (−0.366 − 1.36i)7-s + (0.499 + 0.866i)16-s + (1.36 − 0.366i)19-s − i·25-s + (0.366 − 1.36i)28-s + (1 + i)31-s + (−1.36 − 0.366i)37-s + (−0.866 + 0.5i)49-s + 0.999i·64-s + (−0.366 + 1.36i)67-s + (−1 + i)73-s + (1.36 + 0.366i)76-s + (−1.36 + 0.366i)97-s + (0.5 − 0.866i)100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.970 + 0.240i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.970 + 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.331153992\)
\(L(\frac12)\) \(\approx\) \(1.331153992\)
\(L(1)\) 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 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1 - i)T + iT^{2} \)
37 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.880428291380262164758866036327, −8.695093013214013057024297629476, −7.88433670175675935695324090442, −7.05664712146481997194851636311, −6.76616438476882759252615104054, −5.61173532513933884525156385800, −4.43119338999987451958108208254, −3.53576311710847286512695306089, −2.73496951436913743633077015999, −1.24913713137494876501298609548, 1.55421796591522024873930573635, 2.62876736595995232556270387518, 3.40272903615603880908630079838, 5.01389372601615699602133953016, 5.67487392834577209733013405263, 6.31474192548659897178514497844, 7.22821918520352439886190168963, 8.032596570917918349566136291653, 9.072135549275121375986358079449, 9.650626033726127724541583798004

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