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Properties

Label 2-1512-63.47-c1-0-12
Degree $2$
Conductor $1512$
Sign $0.951 + 0.309i$
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  − 1.36·5-s + (2.64 − 0.0810i)7-s − 1.20i·11-s + (0.639 − 0.369i)13-s + (0.693 + 1.20i)17-s + (2.81 + 1.62i)19-s − 3.81i·23-s − 3.12·25-s + (3.50 + 2.02i)29-s + (1.02 + 0.594i)31-s + (−3.61 + 0.110i)35-s + (5.10 − 8.84i)37-s + (0.670 + 1.16i)41-s + (−0.490 + 0.848i)43-s + (1.63 + 2.83i)47-s + ⋯
L(s)  = 1  − 0.611·5-s + (0.999 − 0.0306i)7-s − 0.362i·11-s + (0.177 − 0.102i)13-s + (0.168 + 0.291i)17-s + (0.646 + 0.373i)19-s − 0.794i·23-s − 0.625·25-s + (0.649 + 0.375i)29-s + (0.184 + 0.106i)31-s + (−0.611 + 0.0187i)35-s + (0.839 − 1.45i)37-s + (0.104 + 0.181i)41-s + (−0.0747 + 0.129i)43-s + (0.238 + 0.413i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.951 + 0.309i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.951 + 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.716142786\)
\(L(\frac12)\) \(\approx\) \(1.716142786\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.0810i)T \)
good5 \( 1 + 1.36T + 5T^{2} \)
11 \( 1 + 1.20iT - 11T^{2} \)
13 \( 1 + (-0.639 + 0.369i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.693 - 1.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.81 - 1.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.81iT - 23T^{2} \)
29 \( 1 + (-3.50 - 2.02i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.02 - 0.594i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.10 + 8.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.670 - 1.16i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.490 - 0.848i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.63 - 2.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.77 + 3.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.73 + 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.36 - 2.51i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.19 - 3.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.84iT - 71T^{2} \)
73 \( 1 + (-10.2 + 5.94i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.34 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.14 + 5.44i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.05 + 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.9 - 6.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.360560190395875796557737554456, −8.408704340963586735946344886807, −7.970971915800147017873439058131, −7.18771122673884726817047194505, −6.10671666160675906968654833216, −5.25546545782729596562756974271, −4.33271085056057359126583619405, −3.52247714860296251084535657322, −2.24654141157593932171325551673, −0.891283674179846381458236894088, 1.07013009422659588338258159658, 2.36689243664885669362765357775, 3.58849326521520568620215581330, 4.51979549346156445239249248336, 5.22440445158525499349094715988, 6.26204697086088052868582653801, 7.37744854164555572252642037692, 7.80341626507758132001415824446, 8.621920904631477910016558961904, 9.489206090151608162581883659012

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