↑

Properties

Label 2-1512-63.47-c1-0-3
Degree $2$
Conductor $1512$
Sign $-0.0594 - 0.998i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  − 2.59·5-s + (−2.61 − 0.417i)7-s − 3.54i·11-s + (2.74 − 1.58i)13-s + (0.487 + 0.844i)17-s + (2.11 + 1.21i)19-s + 3.37i·23-s + 1.72·25-s + (−0.267 − 0.154i)29-s + (−4.35 − 2.51i)31-s + (6.77 + 1.08i)35-s + (−3.47 + 6.01i)37-s + (6.08 + 10.5i)41-s + (−5.47 + 9.48i)43-s + (1.43 + 2.48i)47-s + ⋯
L(s)  = 1  − 1.15·5-s + (−0.987 − 0.157i)7-s − 1.06i·11-s + (0.760 − 0.438i)13-s + (0.118 + 0.204i)17-s + (0.484 + 0.279i)19-s + 0.704i·23-s + 0.345·25-s + (−0.0497 − 0.0286i)29-s + (−0.783 − 0.452i)31-s + (1.14 + 0.183i)35-s + (−0.570 + 0.988i)37-s + (0.949 + 1.64i)41-s + (−0.835 + 1.44i)43-s + (0.209 + 0.362i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0594 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0594 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.0594 - 0.998i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.0594 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6165915528\)
\(L(\frac12)\) \(\approx\) \(0.6165915528\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.61 + 0.417i)T \)
good5 \( 1 + 2.59T + 5T^{2} \)
11 \( 1 + 3.54iT - 11T^{2} \)
13 \( 1 + (-2.74 + 1.58i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.487 - 0.844i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.11 - 1.21i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.37iT - 23T^{2} \)
29 \( 1 + (0.267 + 0.154i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.35 + 2.51i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.47 - 6.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.08 - 10.5i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.47 - 9.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.43 - 2.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.81 - 4.51i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.219 + 0.380i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.41 + 1.97i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.82 - 3.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.25iT - 71T^{2} \)
73 \( 1 + (14.0 - 8.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.49 + 6.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.23 + 12.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.31 - 4.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.4 - 7.20i)T + (48.5 + 84.0i)T^{2} \)
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

−9.667242781354840507557723626272, −8.799519557050053359871616449963, −8.003153027609922868444690081528, −7.46340669004951680481884054927, −6.32083298735555485848452639233, −5.78811623509204586090563656670, −4.48148156771701826624274873182, −3.44176085327090851154595934785, −3.16129442831260525735873387387, −1.09644038722995055581727545349, 0.28678506112182062767018234228, 2.08502781486040648611657605255, 3.42041509309639513880875631158, 3.97458447904642752643041916033, 5.01405264956197570131311867055, 6.07930203297035098968033046202, 7.13268938536909745848239066932, 7.36689899957156280428472812396, 8.628435485383149685357169755578, 9.119956076636166451225286210907

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