↑

Properties

Label 2-735-105.74-c0-0-4
Degree $2$
Conductor $735$
Sign $0.947 - 0.318i$
Analytic cond. $0.366812$
Root an. cond. $0.605650$
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  + (0.707 + 1.22i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 1.41·6-s + (−0.499 − 0.866i)9-s + (0.707 − 1.22i)10-s + (0.5 + 0.866i)12-s − 0.999·15-s + (0.499 + 0.866i)16-s + (0.707 − 1.22i)18-s + (0.707 + 1.22i)19-s + 0.999·20-s + (−0.707 − 1.22i)23-s + (−0.499 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 1.41·6-s + (−0.499 − 0.866i)9-s + (0.707 − 1.22i)10-s + (0.5 + 0.866i)12-s − 0.999·15-s + (0.499 + 0.866i)16-s + (0.707 − 1.22i)18-s + (0.707 + 1.22i)19-s + 0.999·20-s + (−0.707 − 1.22i)23-s + (−0.499 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(0.366812\)
Root analytic conductor: \(0.605650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (704, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :0),\ 0.947 - 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.475760059\)
\(L(\frac12)\) \(\approx\) \(1.475760059\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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

−10.66690550775377775607871865186, −9.386109145748868330507115100820, −8.416642655413968001158965670269, −7.930961220236006144969919157889, −7.17805598239489083324912866829, −6.24878405907328378304464282797, −5.45286664783926690784815793161, −4.40306056601259676609829514075, −3.40871670351277366431031028740, −1.57770252888109236123167016501, 2.17816830613248210525107850738, 3.12498067920438310928304886696, 3.78586094234660544666169587760, 4.66697291079931786673970788632, 5.70368880195604025232033584507, 7.23969834784424516388816539815, 7.957206706603752055599001330593, 9.278851293835554078036192168301, 9.926068045432932154113650683457, 10.72384898474060044634745678799

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