↑

Properties

Label 2-4140-15.8-c1-0-19
Degree $2$
Conductor $4140$
Sign $0.910 - 0.414i$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.54 − 1.62i)5-s + (−0.122 + 0.122i)7-s + 1.17i·11-s + (0.426 + 0.426i)13-s + (1.86 + 1.86i)17-s + 5.98i·19-s + (0.707 − 0.707i)23-s + (−0.252 − 4.99i)25-s + 4.59·29-s − 5.13·31-s + (0.00978 + 0.387i)35-s + (5.60 − 5.60i)37-s + 11.3i·41-s + (1.83 + 1.83i)43-s + (1.48 + 1.48i)47-s + ⋯
L(s)  = 1  + (0.689 − 0.724i)5-s + (−0.0463 + 0.0463i)7-s + 0.353i·11-s + (0.118 + 0.118i)13-s + (0.451 + 0.451i)17-s + 1.37i·19-s + (0.147 − 0.147i)23-s + (−0.0504 − 0.998i)25-s + 0.854·29-s − 0.922·31-s + (0.00165 + 0.0655i)35-s + (0.921 − 0.921i)37-s + 1.76i·41-s + (0.279 + 0.279i)43-s + (0.216 + 0.216i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 0.910 - 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.162790805\)
\(L(\frac12)\) \(\approx\) \(2.162790805\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-1.54 + 1.62i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (0.122 - 0.122i)T - 7iT^{2} \)
11 \( 1 - 1.17iT - 11T^{2} \)
13 \( 1 + (-0.426 - 0.426i)T + 13iT^{2} \)
17 \( 1 + (-1.86 - 1.86i)T + 17iT^{2} \)
19 \( 1 - 5.98iT - 19T^{2} \)
29 \( 1 - 4.59T + 29T^{2} \)
31 \( 1 + 5.13T + 31T^{2} \)
37 \( 1 + (-5.60 + 5.60i)T - 37iT^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 + (-1.83 - 1.83i)T + 43iT^{2} \)
47 \( 1 + (-1.48 - 1.48i)T + 47iT^{2} \)
53 \( 1 + (-1.12 + 1.12i)T - 53iT^{2} \)
59 \( 1 + 2.00T + 59T^{2} \)
61 \( 1 - 2.11T + 61T^{2} \)
67 \( 1 + (3.16 - 3.16i)T - 67iT^{2} \)
71 \( 1 - 5.05iT - 71T^{2} \)
73 \( 1 + (-6.29 - 6.29i)T + 73iT^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 + (8.94 - 8.94i)T - 83iT^{2} \)
89 \( 1 - 6.02T + 89T^{2} \)
97 \( 1 + (10.3 - 10.3i)T - 97iT^{2} \)
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

−8.365595576662392831060175555245, −7.942799689141643364797001800929, −6.97708608695718755461781142700, −6.01238249633669480573453850944, −5.70298221405601878394588174648, −4.67499484806862136920265203166, −4.04379330522994574823452573604, −2.93290721792952746205531446075, −1.88793187432164623321053589081, −1.07055073396163377389930944273, 0.70135302269881519651225987478, 2.03454345041692045143363312048, 2.87038386769889961044252900169, 3.56978475362810037493279353600, 4.72703250516180075080397835867, 5.47483446831717700977393626008, 6.16938932575420092557679046613, 6.97823420287932301852256706932, 7.40858885600053796427439064260, 8.473128959215466748518158380212

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