↑

Properties

Label 2-59e2-59.34-c0-0-1
Degree $2$
Conductor $3481$
Sign $0.0726 - 0.997i$
Analytic cond. $1.73724$
Root an. cond. $1.31804$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.468 − 0.883i)3-s + (0.267 + 0.963i)4-s + (0.725 + 0.687i)5-s + (−0.647 + 0.762i)7-s + (0.725 − 0.687i)12-s + (0.267 − 0.963i)15-s + (−0.856 + 0.515i)16-s + (1.29 + 1.52i)17-s + (−0.907 + 0.419i)19-s + (−0.468 + 0.883i)20-s + (0.976 + 0.214i)21-s + (−0.994 − 0.108i)27-s + (−0.907 − 0.419i)28-s + (−0.796 + 0.605i)29-s + (−0.994 + 0.108i)35-s + ⋯
L(s)  = 1  + (−0.468 − 0.883i)3-s + (0.267 + 0.963i)4-s + (0.725 + 0.687i)5-s + (−0.647 + 0.762i)7-s + (0.725 − 0.687i)12-s + (0.267 − 0.963i)15-s + (−0.856 + 0.515i)16-s + (1.29 + 1.52i)17-s + (−0.907 + 0.419i)19-s + (−0.468 + 0.883i)20-s + (0.976 + 0.214i)21-s + (−0.994 − 0.108i)27-s + (−0.907 − 0.419i)28-s + (−0.796 + 0.605i)29-s + (−0.994 + 0.108i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0726 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0726 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3481\)    =    \(59^{2}\)
Sign: $0.0726 - 0.997i$
Analytic conductor: \(1.73724\)
Root analytic conductor: \(1.31804\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3481} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3481,\ (\ :0),\ 0.0726 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.092714180\)
\(L(\frac12)\) \(\approx\) \(1.092714180\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.267 - 0.963i)T^{2} \)
3 \( 1 + (0.468 + 0.883i)T + (-0.561 + 0.827i)T^{2} \)
5 \( 1 + (-0.725 - 0.687i)T + (0.0541 + 0.998i)T^{2} \)
7 \( 1 + (0.647 - 0.762i)T + (-0.161 - 0.986i)T^{2} \)
11 \( 1 + (-0.468 - 0.883i)T^{2} \)
13 \( 1 + (0.370 - 0.928i)T^{2} \)
17 \( 1 + (-1.29 - 1.52i)T + (-0.161 + 0.986i)T^{2} \)
19 \( 1 + (0.907 - 0.419i)T + (0.647 - 0.762i)T^{2} \)
23 \( 1 + (-0.796 - 0.605i)T^{2} \)
29 \( 1 + (0.796 - 0.605i)T + (0.267 - 0.963i)T^{2} \)
31 \( 1 + (-0.647 - 0.762i)T^{2} \)
37 \( 1 + (0.725 - 0.687i)T^{2} \)
41 \( 1 + (-0.947 + 0.319i)T + (0.796 - 0.605i)T^{2} \)
43 \( 1 + (-0.468 + 0.883i)T^{2} \)
47 \( 1 + (-0.0541 + 0.998i)T^{2} \)
53 \( 1 + (-0.161 + 0.986i)T + (-0.947 - 0.319i)T^{2} \)
61 \( 1 + (-0.267 - 0.963i)T^{2} \)
67 \( 1 + (0.725 + 0.687i)T^{2} \)
71 \( 1 + (1.45 - 1.37i)T + (0.0541 - 0.998i)T^{2} \)
73 \( 1 + (-0.907 + 0.419i)T^{2} \)
79 \( 1 + (0.468 - 0.883i)T + (-0.561 - 0.827i)T^{2} \)
83 \( 1 + (-0.976 + 0.214i)T^{2} \)
89 \( 1 + (-0.267 + 0.963i)T^{2} \)
97 \( 1 + (-0.907 - 0.419i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.781293314576695060230847774372, −8.096074400246728505047163882843, −7.31832478081890141052582700221, −6.67057516194697192579509482739, −6.01806138365211327693690403969, −5.70374951129816491134955915053, −4.06974396834911105822424735763, −3.33333236896805014225671264540, −2.40679666507292013160146969630, −1.64471010475387588141858131694, 0.66374974196621019358646209376, 1.83976419255270197677117602292, 3.04112375889182614402084715889, 4.25429315215759303363371580786, 4.83574600037748723926065665202, 5.57867729858841223736453869314, 6.02483326480208753650557688684, 7.04727379922839525080231601705, 7.66664508219438221973935631977, 9.097186297739881905688736688427

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