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Properties

Label 2-1512-8.5-c1-0-65
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 + 4.29i·5-s − 7-s + (2.28 − 1.66i)8-s + (−5.78 − 1.87i)10-s + 1.60i·11-s − 6.02i·13-s + (0.437 − 1.34i)14-s + (1.23 + 3.80i)16-s − 3.16·17-s − 3.07i·19-s + (5.05 − 6.95i)20-s + (−2.16 − 0.702i)22-s − 5.95·23-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + 1.92i·5-s − 0.377·7-s + (0.809 − 0.587i)8-s + (−1.82 − 0.593i)10-s + 0.484i·11-s − 1.67i·13-s + (0.116 − 0.359i)14-s + (0.309 + 0.951i)16-s − 0.766·17-s − 0.706i·19-s + (1.12 − 1.55i)20-s + (−0.460 − 0.149i)22-s − 1.24·23-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

\(L(1)\) \(\approx\) \(0.1753518938\)
\(L(\frac12)\) \(\approx\) \(0.1753518938\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.29iT - 5T^{2} \)
11 \( 1 - 1.60iT - 11T^{2} \)
13 \( 1 + 6.02iT - 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + 3.07iT - 19T^{2} \)
23 \( 1 + 5.95T + 23T^{2} \)
29 \( 1 + 3.53iT - 29T^{2} \)
31 \( 1 + 2.08T + 31T^{2} \)
37 \( 1 + 4.48iT - 37T^{2} \)
41 \( 1 - 6.69T + 41T^{2} \)
43 \( 1 - 12.2iT - 43T^{2} \)
47 \( 1 + 3.82T + 47T^{2} \)
53 \( 1 + 8.63iT - 53T^{2} \)
59 \( 1 + 1.55iT - 59T^{2} \)
61 \( 1 + 3.64iT - 61T^{2} \)
67 \( 1 + 0.772iT - 67T^{2} \)
71 \( 1 + 7.36T + 71T^{2} \)
73 \( 1 - 1.32T + 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 - 7.08iT - 83T^{2} \)
89 \( 1 + 9.73T + 89T^{2} \)
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

−9.609979094630347332854344359014, −8.207001117463080905470971178615, −7.65462948983889614649824447035, −6.92212033815927397300226040290, −6.26619287578209240458230377469, −5.62368220564291367685036346489, −4.31290627865549592499636509906, −3.29433470196448988691789253123, −2.29963165481286925105366276963, −0.079324578927633981223140124272, 1.31452156709584253939016898229, 2.15296424493083243212414703735, 3.82913798590423870642517048590, 4.29470054359221233898973739638, 5.19380912084870270378617283828, 6.17463168042542256662390592484, 7.50202087826393972304035988305, 8.444668159480492580122562514825, 8.938111957103843778829775298358, 9.401212662362531275878755325154

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