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Properties

Label 2-1539-171.131-c0-0-0
Degree $2$
Conductor $1539$
Sign $0.630 + 0.776i$
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  + (0.766 − 0.642i)4-s + (0.939 − 1.62i)7-s + (0.266 + 1.50i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)19-s + (−0.939 − 0.342i)25-s + (−0.326 − 1.85i)28-s + 0.347·31-s − 1.87·37-s + (0.266 + 0.223i)43-s + (−1.26 − 2.19i)49-s + (1.17 + 0.984i)52-s + (0.266 + 1.50i)61-s + (−0.500 − 0.866i)64-s + (1.76 + 0.642i)67-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)4-s + (0.939 − 1.62i)7-s + (0.266 + 1.50i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)19-s + (−0.939 − 0.342i)25-s + (−0.326 − 1.85i)28-s + 0.347·31-s − 1.87·37-s + (0.266 + 0.223i)43-s + (−1.26 − 2.19i)49-s + (1.17 + 0.984i)52-s + (0.266 + 1.50i)61-s + (−0.500 − 0.866i)64-s + (1.76 + 0.642i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1539\)    =    \(3^{4} \cdot 19\)
Sign: $0.630 + 0.776i$
Analytic conductor: \(0.768061\)
Root analytic conductor: \(0.876390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1539} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1539,\ (\ :0),\ 0.630 + 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.429860829\)
\(L(\frac12)\) \(\approx\) \(1.429860829\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.766 + 0.642i)T^{2} \)
5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 - 0.347T + T^{2} \)
37 \( 1 + 1.87T + T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.839036635875506689744343577677, −8.650382044158021148447352052988, −7.79453633842501404835591115524, −7.03049023070501881955062667128, −6.50603252113923799294352320884, −5.42865492090309232827935010196, −4.40047188461384447810043274469, −3.76338598162392126990509651957, −2.07289366321321238224563888294, −1.32029729186612737803445188890, 1.86958807628796803968049289906, 2.65381813564669162668896695511, 3.57697497779721537029569596983, 4.99508638760062759611069931061, 5.63030651034592048352119998258, 6.47925228368821255103646689308, 7.51927962535623997856574907179, 8.305065138818329809778676915784, 8.607771762043015982129751714150, 9.692901482578110050797919721123

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