Properties

Label 2-3528-72.5-c0-0-2
Degree $2$
Conductor $3528$
Sign $-0.342 + 0.939i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
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  + (0.866 − 0.5i)2-s + (−0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.258 + 0.448i)5-s + (−0.258 + 0.965i)6-s − 0.999i·8-s − 1.00i·9-s + 0.517i·10-s + (0.258 + 0.965i)12-s + (−1.22 − 0.707i)13-s + (−0.133 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s − 1.93i·19-s + (0.258 + 0.448i)20-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.258 + 0.448i)5-s + (−0.258 + 0.965i)6-s − 0.999i·8-s − 1.00i·9-s + 0.517i·10-s + (0.258 + 0.965i)12-s + (−1.22 − 0.707i)13-s + (−0.133 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s − 1.93i·19-s + (0.258 + 0.448i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.105070558\)
\(L(\frac12)\) \(\approx\) \(1.105070558\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 \)
good5 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.93iT - T^{2} \)
23 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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

−8.716256959358050996044721806960, −7.46776965118387026457418347325, −6.82633536465333308237746538402, −6.14429272205118999563739406814, −5.18590578769867613105222848073, −4.80342716109298904117059801697, −3.93676095084480345233941139564, −3.06936611156544679061754938173, −2.29569236629904224395054412134, −0.49792725279795180731054239591, 1.65096595252333402814803175165, 2.49482337431768387486689217039, 3.88084646101948742872495817432, 4.45150693192377754555810129274, 5.41240329347475066091063475220, 5.84348428060557406341314516022, 6.70272717068835276734044368302, 7.37686226899432917003102426641, 7.977372074878969509952966538066, 8.547964202662124504714288854621

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