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Properties

Label 2-1512-63.25-c1-0-19
Degree $2$
Conductor $1512$
Sign $-0.386 + 0.922i$
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  + (1.92 − 3.32i)5-s + (−2.55 + 0.693i)7-s + (0.903 + 1.56i)11-s + (−0.692 − 1.19i)13-s + (0.833 − 1.44i)17-s + (−0.0802 − 0.138i)19-s + (1.60 − 2.77i)23-s + (−4.87 − 8.44i)25-s + (3.78 − 6.54i)29-s + 3.22·31-s + (−2.59 + 9.82i)35-s + (1.58 + 2.74i)37-s + (−6.00 − 10.3i)41-s + (3.45 − 5.98i)43-s − 11.4·47-s + ⋯
L(s)  = 1  + (0.858 − 1.48i)5-s + (−0.965 + 0.261i)7-s + (0.272 + 0.471i)11-s + (−0.192 − 0.332i)13-s + (0.202 − 0.350i)17-s + (−0.0184 − 0.0318i)19-s + (0.333 − 0.577i)23-s + (−0.975 − 1.68i)25-s + (0.701 − 1.21i)29-s + 0.578·31-s + (−0.439 + 1.66i)35-s + (0.260 + 0.451i)37-s + (−0.937 − 1.62i)41-s + (0.526 − 0.912i)43-s − 1.66·47-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: $-0.386 + 0.922i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.459324973\)
\(L(\frac12)\) \(\approx\) \(1.459324973\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.55 - 0.693i)T \)
good5 \( 1 + (-1.92 + 3.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.903 - 1.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.692 + 1.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.833 + 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0802 + 0.138i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.60 + 2.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.78 + 6.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.22T + 31T^{2} \)
37 \( 1 + (-1.58 - 2.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.00 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.45 + 5.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + (1.37 - 2.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 15.0T + 59T^{2} \)
61 \( 1 + 9.20T + 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 - 6.93T + 71T^{2} \)
73 \( 1 + (6.22 - 10.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 16.0T + 79T^{2} \)
83 \( 1 + (-1.45 + 2.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.04 + 8.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.18 + 7.25i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.207634633425985723822115089736, −8.687173747881565000685987858049, −7.72867545773118373830351041307, −6.58194471886597470314283416109, −5.93124851197665545729464320730, −5.05073246617238083381260949008, −4.35688402356031894272149777577, −3.00163422598269068355618057264, −1.88549778146396237599500997835, −0.56647414220612905975088640566, 1.59994396438248432859177557161, 3.03842219862413385231968605726, 3.26444009552693776902076287828, 4.72981265341618504171280345953, 6.05005665270341762876034467013, 6.38876833172687784164339229044, 7.07699968331694753322128529445, 8.007216203754678213678428244967, 9.275244115341902466374725018332, 9.687313898305027143984978959900

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