Properties

Degree $2$
Conductor $1512$
Sign $-0.239 + 0.970i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0144 − 0.0250i)5-s + (−1.70 + 2.02i)7-s + (−1.88 + 3.25i)11-s − 2.97·13-s + (0.708 − 1.22i)17-s + (−3.81 − 6.60i)19-s + (−2.98 − 5.17i)23-s + (2.49 − 4.32i)25-s + 9.99·29-s + (4.27 − 7.40i)31-s + (0.0753 + 0.0135i)35-s + (−0.737 − 1.27i)37-s − 4.38·41-s + 3.76·43-s + (2.07 + 3.59i)47-s + ⋯
L(s)  = 1  + (−0.00646 − 0.0112i)5-s + (−0.645 + 0.763i)7-s + (−0.567 + 0.982i)11-s − 0.824·13-s + (0.171 − 0.297i)17-s + (−0.874 − 1.51i)19-s + (−0.622 − 1.07i)23-s + (0.499 − 0.865i)25-s + 1.85·29-s + (0.768 − 1.33i)31-s + (0.0127 + 0.00229i)35-s + (−0.121 − 0.209i)37-s − 0.685·41-s + 0.573·43-s + (0.302 + 0.524i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.239 + 0.970i$
Motivic weight: \(1\)
Character: $\chi_{1512} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.239 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7024166361\)
\(L(\frac12)\) \(\approx\) \(0.7024166361\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 2.02i)T \)
good5 \( 1 + (0.0144 + 0.0250i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.88 - 3.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.97T + 13T^{2} \)
17 \( 1 + (-0.708 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.81 + 6.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.98 + 5.17i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.99T + 29T^{2} \)
31 \( 1 + (-4.27 + 7.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.737 + 1.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.38T + 41T^{2} \)
43 \( 1 - 3.76T + 43T^{2} \)
47 \( 1 + (-2.07 - 3.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.60 + 6.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.93 - 12.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.61 + 9.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.77 - 4.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + (-2.17 + 3.77i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.04 + 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.62T + 83T^{2} \)
89 \( 1 + (-2.14 - 3.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.77T + 97T^{2} \)
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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

−9.218245092407488329774714925489, −8.548657793898941291165589276889, −7.62869434108987821757221214200, −6.71117452091730277550745144368, −6.13913933520424565996692733957, −4.84410260709618203824389885304, −4.46295336567854057738255835915, −2.69159219257094861740908904828, −2.43304984964055521537650213481, −0.28054824664493577318026899242, 1.31557516800717089364775772029, 2.87355851524272039968186743673, 3.61751386164799039522143809451, 4.65825207660471633133634083673, 5.69686141357634328260645945207, 6.43389669265063559214339463788, 7.32842236175847533557103403799, 8.105900564118103403668053887607, 8.807397890237589335116236862519, 10.00480537210211636521475022330

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