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Properties

Label 2-1512-63.4-c1-0-14
Degree $2$
Conductor $1512$
Sign $-0.0788 + 0.996i$
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  − 5-s + (−2.5 + 0.866i)7-s + 3·11-s + (−0.5 + 0.866i)13-s + (1.5 − 2.59i)17-s + (−2.5 − 4.33i)19-s − 23-s − 4·25-s + (4.5 + 7.79i)29-s + (−2 − 3.46i)31-s + (2.5 − 0.866i)35-s + (−2.5 − 4.33i)37-s + (3.5 − 6.06i)41-s + (−1.5 − 2.59i)43-s + (4 − 6.92i)47-s + ⋯
L(s)  = 1  − 0.447·5-s + (−0.944 + 0.327i)7-s + 0.904·11-s + (−0.138 + 0.240i)13-s + (0.363 − 0.630i)17-s + (−0.573 − 0.993i)19-s − 0.208·23-s − 0.800·25-s + (0.835 + 1.44i)29-s + (−0.359 − 0.622i)31-s + (0.422 − 0.146i)35-s + (−0.410 − 0.711i)37-s + (0.546 − 0.946i)41-s + (−0.228 − 0.396i)43-s + (0.583 − 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.0788 + 0.996i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.0788 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8872769089\)
\(L(\frac12)\) \(\approx\) \(0.8872769089\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.5 + 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.5 + 2.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.5 + 11.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.079544402297958354006976135980, −8.782469863475286379652677179194, −7.47392331068586121008046230826, −6.88858380727272206122658968619, −6.09797342313201400398079315702, −5.10318118359659614491679834001, −4.03207950475789621943506294131, −3.27579745282555623266205744600, −2.10393135698600217566451032853, −0.37518655202211718665225645374, 1.27103909602568448564728803852, 2.79283942078935197655726175668, 3.83820997555562842987112505327, 4.33940064343357763927277691063, 5.88928360775376908759111551739, 6.29866622011716750690264289548, 7.31566825169507251832636232699, 8.063683132379836087955773374077, 8.857707793223250808755212040489, 9.881270935275064878716957596930

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