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Properties

Label 2-1539-171.103-c0-0-0
Degree $2$
Conductor $1539$
Sign $0.714 + 0.699i$
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 − 1.73i·13-s + 16-s − 19-s + (0.5 − 0.866i)25-s + (−0.5 − 0.866i)28-s + (1.5 + 0.866i)31-s + 1.73i·37-s − 43-s − 1.73i·52-s + (−0.5 + 0.866i)61-s + 64-s + 1.73i·67-s + (0.5 − 0.866i)73-s + ⋯
L(s)  = 1  + 4-s + (−0.5 − 0.866i)7-s − 1.73i·13-s + 16-s − 19-s + (0.5 − 0.866i)25-s + (−0.5 − 0.866i)28-s + (1.5 + 0.866i)31-s + 1.73i·37-s − 43-s − 1.73i·52-s + (−0.5 + 0.866i)61-s + 64-s + 1.73i·67-s + (0.5 − 0.866i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1539\)    =    \(3^{4} \cdot 19\)
Sign: $0.714 + 0.699i$
Analytic conductor: \(0.768061\)
Root analytic conductor: \(0.876390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1539} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1539,\ (\ :0),\ 0.714 + 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.304811610\)
\(L(\frac12)\) \(\approx\) \(1.304811610\)
\(L(1)\) 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 \( 1 + 1.73iT - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
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 \( 1 - 1.73iT - T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.00597349039159117736806707201, −8.413279572420646392097483091414, −8.049868133724637693648956409674, −6.93362892178194455364952346197, −6.52840832156234312111633446302, −5.57984844451973226026706801074, −4.48689724181073750260162206022, −3.29922511709881655674458067753, −2.65761141777823131694839990634, −1.09131563528078784096830501243, 1.81714122993296504725738213255, 2.55005661942751767719644105751, 3.67601876674228284170968734201, 4.76841214681768873303051112720, 5.95804595716278185437742045715, 6.46248177996568098345760344533, 7.16000461105865088925132869119, 8.158801950339140063273421195843, 9.062450808790791101198532086064, 9.601368254141669842125799212382

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