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Properties

Label 2-1539-171.68-c0-0-0
Degree $2$
Conductor $1539$
Sign $0.925 - 0.377i$
Analytic cond. $0.768061$
Root an. cond. $0.876390$
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  + 4-s + (0.5 + 0.866i)7-s − 13-s + 16-s + 19-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)28-s + (0.5 − 0.866i)31-s − 37-s − 43-s − 52-s + (0.5 − 0.866i)61-s + 64-s − 67-s + (0.5 − 0.866i)73-s + ⋯
L(s)  = 1  + 4-s + (0.5 + 0.866i)7-s − 13-s + 16-s + 19-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)28-s + (0.5 − 0.866i)31-s − 37-s − 43-s − 52-s + (0.5 − 0.866i)61-s + 64-s − 67-s + (0.5 − 0.866i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1539\)    =    \(3^{4} \cdot 19\)
Sign: $0.925 - 0.377i$
Analytic conductor: \(0.768061\)
Root analytic conductor: \(0.876390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1539} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1539,\ (\ :0),\ 0.925 - 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.436261026\)
\(L(\frac12)\) \(\approx\) \(1.436261026\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 - T \)
good2 \( 1 - T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−9.756974307907275416324768914426, −8.911818576233452463272809483768, −7.906277925318936395148503235966, −7.39170794713973624650260959775, −6.47987503532712501661389572456, −5.55592948597905320287143075294, −4.96197525067685487367701666691, −3.50818336541310667620523576584, −2.55077014801167025377895925272, −1.68354271794708447845707629030, 1.33439010228231421956922834015, 2.49616759794252448577206942042, 3.50210846922613935971651846994, 4.63151513257975307605923674579, 5.47842805358497058530141016389, 6.54878749917281417990758012606, 7.25727411380110451526731399073, 7.75709023521877443322407063983, 8.676376175225585669733257482283, 9.957234692703789338578407344723

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