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Properties

Label 2-1539-171.31-c0-0-0
Degree $2$
Conductor $1539$
Sign $-0.432 + 0.901i$
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  + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)7-s + (1.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s − 19-s − 25-s + (−0.499 + 0.866i)28-s + (−1.5 − 0.866i)31-s − 1.73i·37-s + (0.5 − 0.866i)43-s + (−1.5 − 0.866i)52-s + 61-s + 0.999·64-s + (−1.5 + 0.866i)67-s + (0.5 + 0.866i)73-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)7-s + (1.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s − 19-s − 25-s + (−0.499 + 0.866i)28-s + (−1.5 − 0.866i)31-s − 1.73i·37-s + (0.5 − 0.866i)43-s + (−1.5 − 0.866i)52-s + 61-s + 0.999·64-s + (−1.5 + 0.866i)67-s + (0.5 + 0.866i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1539\)    =    \(3^{4} \cdot 19\)
Sign: $-0.432 + 0.901i$
Analytic conductor: \(0.768061\)
Root analytic conductor: \(0.876390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1539} (1000, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1539,\ (\ :0),\ -0.432 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7946910073\)
\(L(\frac12)\) \(\approx\) \(0.7946910073\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.391674747827343721779276836632, −8.740171492050629925201440756384, −7.83853812601209428190800578214, −6.88196841585388463797405515597, −5.95169132394668030626921731926, −5.50747404749149850331893850695, −4.07377973894524605089867667897, −3.73686595431700663524881313362, −2.02538567623731203583895114242, −0.64519164497208465270676077102, 1.88452417706379162889654661512, 3.14582775650617736141780192693, 3.87694072167989860973710595584, 4.79732564778030276282378954499, 5.98008220270102368929645240833, 6.55952016578457544114604140707, 7.64283768249470083613452075369, 8.533457250131720265111371547686, 8.940263913721101997530071213278, 9.619975017384869277746101429845

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