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Properties

Label 2-1512-63.4-c1-0-22
Degree $2$
Conductor $1512$
Sign $-0.0895 + 0.995i$
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  + 3.43·5-s + (−1.83 − 1.90i)7-s − 4.40·11-s + (1.49 − 2.58i)13-s + (−0.542 + 0.939i)17-s + (−3.74 − 6.48i)19-s + 4.32·23-s + 6.80·25-s + (−1.68 − 2.91i)29-s + (−4.68 − 8.11i)31-s + (−6.31 − 6.53i)35-s + (−2.50 − 4.34i)37-s + (1.20 − 2.08i)41-s + (3.31 + 5.74i)43-s + (1.50 − 2.60i)47-s + ⋯
L(s)  = 1  + 1.53·5-s + (−0.695 − 0.718i)7-s − 1.32·11-s + (0.414 − 0.717i)13-s + (−0.131 + 0.227i)17-s + (−0.858 − 1.48i)19-s + 0.901·23-s + 1.36·25-s + (−0.312 − 0.541i)29-s + (−0.841 − 1.45i)31-s + (−1.06 − 1.10i)35-s + (−0.412 − 0.714i)37-s + (0.187 − 0.325i)41-s + (0.505 + 0.875i)43-s + (0.219 − 0.380i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0895 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0895 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.0895 + 0.995i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.0895 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.557880778\)
\(L(\frac12)\) \(\approx\) \(1.557880778\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 + 1.90i)T \)
good5 \( 1 - 3.43T + 5T^{2} \)
11 \( 1 + 4.40T + 11T^{2} \)
13 \( 1 + (-1.49 + 2.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.542 - 0.939i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.74 + 6.48i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.32T + 23T^{2} \)
29 \( 1 + (1.68 + 2.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.68 + 8.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.50 + 4.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.20 + 2.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.31 - 5.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.50 + 2.60i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.530 + 0.919i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.20 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.71 + 4.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.66 + 2.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (8.21 - 14.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.17 + 2.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.60 - 2.78i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.67 + 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.40 + 11.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.403321338306766454940217981002, −8.621258662240519086301924500895, −7.53529635705665409575695077399, −6.78555867499632275404642783487, −5.88133778102972628826294432737, −5.35351634799814732436514322844, −4.21529129879808500791479454716, −2.90301326474503868591842474715, −2.19890220856404329168533849023, −0.57080016673159066003698700218, 1.68685096769539461918199606858, 2.48864609947763235863578447024, 3.49885394329722571810058677074, 5.02675669965941249541466546211, 5.59150005049058278237031250972, 6.32571962711482676627514960940, 7.04919590067918605090159536859, 8.331044981690907486923572249530, 8.990589043999004662666057756858, 9.650270434493762241411829550804

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