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Properties

Label 2-1512-63.16-c1-0-18
Degree $2$
Conductor $1512$
Sign $-0.771 + 0.636i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.77·5-s + (0.855 − 2.50i)7-s + 3.43·11-s + (−0.429 − 0.743i)13-s + (0.405 + 0.701i)17-s + (0.750 − 1.29i)19-s − 7.64·23-s + 2.68·25-s + (−3.99 + 6.92i)29-s + (3.60 − 6.24i)31-s + (−2.37 + 6.93i)35-s + (0.458 − 0.793i)37-s + (−1.67 − 2.90i)41-s + (1.20 − 2.08i)43-s + (−0.307 − 0.532i)47-s + ⋯
L(s)  = 1  − 1.23·5-s + (0.323 − 0.946i)7-s + 1.03·11-s + (−0.119 − 0.206i)13-s + (0.0982 + 0.170i)17-s + (0.172 − 0.298i)19-s − 1.59·23-s + 0.536·25-s + (−0.742 + 1.28i)29-s + (0.647 − 1.12i)31-s + (−0.400 + 1.17i)35-s + (0.0753 − 0.130i)37-s + (−0.261 − 0.453i)41-s + (0.183 − 0.318i)43-s + (−0.0448 − 0.0776i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.771 + 0.636i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.771 + 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6943738589\)
\(L(\frac12)\) \(\approx\) \(0.6943738589\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.855 + 2.50i)T \)
good5 \( 1 + 2.77T + 5T^{2} \)
11 \( 1 - 3.43T + 11T^{2} \)
13 \( 1 + (0.429 + 0.743i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.405 - 0.701i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.750 + 1.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.64T + 23T^{2} \)
29 \( 1 + (3.99 - 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.60 + 6.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.458 + 0.793i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.67 + 2.90i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.20 + 2.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.307 + 0.532i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.31 + 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.734 + 1.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.71 + 9.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.10 - 14.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (4.16 + 7.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.37 - 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.75 + 9.97i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.11 + 8.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.82 + 6.63i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.096948682106048253356835412496, −8.176207757833582373105038558197, −7.61107721419777784667784811882, −6.93684498522998552795965186310, −5.96614943718274270724040725858, −4.69475199221694731285785504700, −3.99890993808702821659865255287, −3.39152055664317706220614309799, −1.68567845840308671785710877009, −0.28251602250033796718001700160, 1.54915141275242103065380355926, 2.87499964342144677143653125880, 3.96113283407428414798742466100, 4.56005137186555250014317091801, 5.78444834284872365121631997384, 6.48383511464737133205967892496, 7.64846425875397311124208649032, 8.060021332826828852161319790214, 8.957948767904428876886101066383, 9.605554497318809202694610508153

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