↑

Properties

Label 2-756-63.41-c1-0-3
Degree $2$
Conductor $756$
Sign $0.967 - 0.254i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (0.266 − 0.462i)5-s + (−1.89 + 1.84i)7-s + (3.39 − 1.96i)11-s + (0.116 + 0.0674i)13-s + 4.32·17-s + 2.22i·19-s + (1.70 + 0.983i)23-s + (2.35 + 4.08i)25-s + (5.16 − 2.98i)29-s + (0.800 + 0.462i)31-s + (0.348 + 1.36i)35-s + 7.79·37-s + (−4.59 + 7.95i)41-s + (3.24 + 5.62i)43-s + (−3.04 − 5.27i)47-s + ⋯
L(s)  = 1  + (0.119 − 0.206i)5-s + (−0.715 + 0.698i)7-s + (1.02 − 0.591i)11-s + (0.0324 + 0.0187i)13-s + 1.04·17-s + 0.511i·19-s + (0.355 + 0.205i)23-s + (0.471 + 0.816i)25-s + (0.959 − 0.553i)29-s + (0.143 + 0.0829i)31-s + (0.0589 + 0.231i)35-s + 1.28·37-s + (−0.716 + 1.24i)41-s + (0.494 + 0.857i)43-s + (−0.443 − 0.768i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.967 - 0.254i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.967 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55665 + 0.201332i\)
\(L(\frac12)\) \(\approx\) \(1.55665 + 0.201332i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (1.89 - 1.84i)T \)
good5 \( 1 + (-0.266 + 0.462i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.39 + 1.96i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.116 - 0.0674i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 - 2.22iT - 19T^{2} \)
23 \( 1 + (-1.70 - 0.983i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.16 + 2.98i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.800 - 0.462i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.79T + 37T^{2} \)
41 \( 1 + (4.59 - 7.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.24 - 5.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.04 + 5.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.0iT - 53T^{2} \)
59 \( 1 + (1.89 - 3.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.35 + 5.39i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.75 - 9.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.22iT - 71T^{2} \)
73 \( 1 - 0.381iT - 73T^{2} \)
79 \( 1 + (4.60 + 7.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.28 + 2.21i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 17.1T + 89T^{2} \)
97 \( 1 + (13.6 - 7.89i)T + (48.5 - 84.0i)T^{2} \)
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

−10.14230289874141098225951194748, −9.547252268765113303166280748707, −8.749279454974083060806296460888, −7.936087499158450691729071837885, −6.67984848880834206711172176569, −6.03453442800117474397523456939, −5.10140056101265380383167525516, −3.75639611243303061816317556220, −2.86480121332097655062777523518, −1.21726891309032144063351924398, 1.03898342114966714446696765034, 2.73704627157288639363484620207, 3.81650683205556723569941520960, 4.75508964885732388830180092220, 6.09182670471569536013012571458, 6.81740220152873559779765916089, 7.53529254940956152899305321321, 8.729725010434962505499992837874, 9.560767770331334519749385652126, 10.24917784409560566283510906155

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