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Properties

Label 2-1512-8.5-c1-0-47
Degree $2$
Conductor $1512$
Sign $0.587 - 0.809i$
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  + (0.437 + 1.34i)2-s + (−1.61 + 1.17i)4-s + 0.0549i·5-s − 7-s + (−2.28 − 1.66i)8-s + (−0.0738 + 0.0240i)10-s − 2.63i·11-s − 3.67i·13-s + (−0.437 − 1.34i)14-s + (1.23 − 3.80i)16-s + 3.16·17-s + 3.07i·19-s + (−0.0645 − 0.0888i)20-s + (3.54 − 1.15i)22-s + 2.86·23-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + 0.0245i·5-s − 0.377·7-s + (−0.809 − 0.587i)8-s + (−0.0233 + 0.00759i)10-s − 0.794i·11-s − 1.02i·13-s + (−0.116 − 0.359i)14-s + (0.309 − 0.951i)16-s + 0.766·17-s + 0.706i·19-s + (−0.0144 − 0.0198i)20-s + (0.755 − 0.245i)22-s + 0.598·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.587 - 0.809i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.587 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.701731608\)
\(L(\frac12)\) \(\approx\) \(1.701731608\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.437 - 1.34i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 0.0549iT - 5T^{2} \)
11 \( 1 + 2.63iT - 11T^{2} \)
13 \( 1 + 3.67iT - 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 - 3.07iT - 19T^{2} \)
23 \( 1 - 2.86T + 23T^{2} \)
29 \( 1 - 10.1iT - 29T^{2} \)
31 \( 1 - 9.32T + 31T^{2} \)
37 \( 1 - 0.774iT - 37T^{2} \)
41 \( 1 - 6.36T + 41T^{2} \)
43 \( 1 + 9.98iT - 43T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 + 3.39iT - 53T^{2} \)
59 \( 1 - 6.93iT - 59T^{2} \)
61 \( 1 + 8.35iT - 61T^{2} \)
67 \( 1 + 8.93iT - 67T^{2} \)
71 \( 1 - 4.28T + 71T^{2} \)
73 \( 1 - 8.38T + 73T^{2} \)
79 \( 1 + 3.03T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−9.428764820268629316800996680285, −8.504611652824432376027935455751, −8.053959539824666996910696261815, −7.06768115469402476106420682175, −6.35034382248634010523893517582, −5.50250424900363193963868755095, −4.87793900722783806721621708671, −3.50289494768508985086582448231, −3.04596820336870146181603422033, −0.840458329089072446313635442260, 0.984182298639274236311837002077, 2.29938419811055601679950593633, 3.13542748113610218729460100381, 4.37058142461529351632430625159, 4.78651842745567215573923251360, 6.03861376800382628510186558845, 6.76019899235018986511216174082, 7.88242259920585958887159117367, 8.821412512704574142005708939392, 9.702884955907390648057477994629

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