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Properties

Label 2-1512-63.47-c1-0-0
Degree $2$
Conductor $1512$
Sign $-0.900 - 0.434i$
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  + 0.207·5-s + (−1.37 − 2.26i)7-s + 1.51i·11-s + (−3.39 + 1.95i)13-s + (0.873 + 1.51i)17-s + (−0.968 − 0.559i)19-s − 1.44i·23-s − 4.95·25-s + (−1.99 − 1.15i)29-s + (−5.15 − 2.97i)31-s + (−0.283 − 0.468i)35-s + (−2.13 + 3.69i)37-s + (2.91 + 5.04i)41-s + (−0.213 + 0.369i)43-s + (4.13 + 7.16i)47-s + ⋯
L(s)  = 1  + 0.0925·5-s + (−0.518 − 0.855i)7-s + 0.457i·11-s + (−0.940 + 0.543i)13-s + (0.211 + 0.366i)17-s + (−0.222 − 0.128i)19-s − 0.301i·23-s − 0.991·25-s + (−0.370 − 0.214i)29-s + (−0.926 − 0.534i)31-s + (−0.0479 − 0.0791i)35-s + (−0.351 + 0.608i)37-s + (0.454 + 0.787i)41-s + (−0.0325 + 0.0563i)43-s + (0.603 + 1.04i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.900 - 0.434i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.900 - 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2255637836\)
\(L(\frac12)\) \(\approx\) \(0.2255637836\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.37 + 2.26i)T \)
good5 \( 1 - 0.207T + 5T^{2} \)
11 \( 1 - 1.51iT - 11T^{2} \)
13 \( 1 + (3.39 - 1.95i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.873 - 1.51i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.968 + 0.559i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.44iT - 23T^{2} \)
29 \( 1 + (1.99 + 1.15i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.15 + 2.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.13 - 3.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.91 - 5.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.213 - 0.369i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.13 - 7.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.16 - 4.71i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.63 - 2.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.5 - 6.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.24 + 5.62i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.10iT - 71T^{2} \)
73 \( 1 + (-10.6 + 6.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.57 + 4.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.36 - 14.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.96 - 3.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.1 + 7.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.592430935712951577628425954060, −9.400747606548387272898885164224, −8.013056320073953911405765394322, −7.43462074652230919562846822025, −6.63928418295648178835224214803, −5.82474946929037234368031226280, −4.63212255885107314633479928454, −4.00529729282240072317885866320, −2.82983990947755204971065613732, −1.63022682708150806109322036581, 0.082980994384256497312121583012, 1.96360063821105152439765662863, 2.95495647560269336117357379911, 3.87117559578798258989826632991, 5.28519299446199741250431996990, 5.62882375538971379019586103275, 6.70629494214018927086725669280, 7.53284191016892791909751686947, 8.379579511463067309881779101422, 9.247300640315756840426473916984

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