Properties

Degree $2$
Conductor $1512$
Sign $0.386 + 0.922i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 3.46i)5-s + (2 + 1.73i)7-s + (−1 − 1.73i)11-s + 5·13-s + (−3 − 5.19i)17-s + (2 − 3.46i)19-s + (−3 + 5.19i)23-s + (−5.49 − 9.52i)25-s + 6·29-s + (3.5 + 6.06i)31-s + (10 − 3.46i)35-s + (−3.5 + 6.06i)37-s + 2·41-s − 7·43-s + (1 − 1.73i)47-s + ⋯
L(s)  = 1  + (0.894 − 1.54i)5-s + (0.755 + 0.654i)7-s + (−0.301 − 0.522i)11-s + 1.38·13-s + (−0.727 − 1.26i)17-s + (0.458 − 0.794i)19-s + (−0.625 + 1.08i)23-s + (−1.09 − 1.90i)25-s + 1.11·29-s + (0.628 + 1.08i)31-s + (1.69 − 0.585i)35-s + (−0.575 + 0.996i)37-s + 0.312·41-s − 1.06·43-s + (0.145 − 0.252i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.386 + 0.922i$
Motivic weight: \(1\)
Character: $\chi_{1512} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.217176138\)
\(L(\frac12)\) \(\approx\) \(2.217176138\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15T + 97T^{2} \)
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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

−9.188433984246528729469153003989, −8.508385544461882300850942521957, −8.196760255591124082067706619074, −6.73858085658448579324079350761, −5.84135779008309579375284520123, −5.09278638966737615585640766534, −4.67041488364279331821365587351, −3.14538576831699884524775159952, −1.87784844181740957651219037248, −0.955945925061884059852400536299, 1.55869997595917107007756803860, 2.43833320478230488071768253303, 3.63483560584070024617651022378, 4.43713399790362895547859330853, 5.87286814712076321713532441905, 6.27925044411713887298499256319, 7.12570850932302715700973426988, 7.984971492146270062353283586741, 8.720055336694627882912932059129, 10.01700425014403252979344240697

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