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Properties

Label 2-1512-63.47-c1-0-18
Degree $2$
Conductor $1512$
Sign $-0.0988 + 0.995i$
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  − 2.20·5-s + (2.16 − 1.52i)7-s − 0.181i·11-s + (2.50 − 1.44i)13-s + (−1.98 − 3.43i)17-s + (0.867 + 0.500i)19-s + 5.61i·23-s − 0.121·25-s + (−0.703 − 0.406i)29-s + (−6.89 − 3.98i)31-s + (−4.78 + 3.35i)35-s + (1.25 − 2.17i)37-s + (0.612 + 1.06i)41-s + (5.47 − 9.48i)43-s + (−3.57 − 6.19i)47-s + ⋯
L(s)  = 1  − 0.987·5-s + (0.818 − 0.574i)7-s − 0.0548i·11-s + (0.694 − 0.400i)13-s + (−0.481 − 0.833i)17-s + (0.198 + 0.114i)19-s + 1.17i·23-s − 0.0242·25-s + (−0.130 − 0.0754i)29-s + (−1.23 − 0.715i)31-s + (−0.808 + 0.567i)35-s + (0.206 − 0.357i)37-s + (0.0957 + 0.165i)41-s + (0.835 − 1.44i)43-s + (−0.521 − 0.903i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0988 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0988 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.0988 + 0.995i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.0988 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.171950299\)
\(L(\frac12)\) \(\approx\) \(1.171950299\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 + 1.52i)T \)
good5 \( 1 + 2.20T + 5T^{2} \)
11 \( 1 + 0.181iT - 11T^{2} \)
13 \( 1 + (-2.50 + 1.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.98 + 3.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.867 - 0.500i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.61iT - 23T^{2} \)
29 \( 1 + (0.703 + 0.406i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.89 + 3.98i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.25 + 2.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.612 - 1.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.47 + 9.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.57 + 6.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.75 + 1.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.27 + 5.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.97 + 4.02i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.44 - 5.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (10.1 - 5.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.35 + 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.19 + 12.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.11 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.01 + 1.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.127532325202570258106300544161, −8.383211775139525203648132601467, −7.49985953334548205755253718494, −7.26019845454689907061592589859, −5.88165494164299984249539515913, −5.03827373490704597179552368092, −4.03391073179087939256196897055, −3.45298193369134142789146750668, −1.91788737195909770191235104610, −0.49267339110820771738615781103, 1.39292224034462326139616089177, 2.64129312546539467674533391782, 3.90948849482509825259406895103, 4.49134986740905148293161432331, 5.55926067355953173952106308152, 6.46235220647144062286568986843, 7.39606779032125300814230619812, 8.245254844202475098131547776801, 8.641915951277896797493057639435, 9.534314394951435586842942459079

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