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Properties

Label 2-1512-8.5-c1-0-86
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  + (−1.14 + 0.831i)2-s + (0.618 − 1.90i)4-s − 2.63i·5-s − 7-s + (0.874 + 2.68i)8-s + (2.19 + 3.01i)10-s − 4.29i·11-s − 0.0480i·13-s + (1.14 − 0.831i)14-s + (−3.23 − 2.35i)16-s − 3.16·17-s − 0.726i·19-s + (−5.01 − 1.62i)20-s + (3.57 + 4.91i)22-s − 5.28·23-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s − 1.17i·5-s − 0.377·7-s + (0.309 + 0.951i)8-s + (0.692 + 0.953i)10-s − 1.29i·11-s − 0.0133i·13-s + (0.305 − 0.222i)14-s + (−0.809 − 0.587i)16-s − 0.766·17-s − 0.166i·19-s + (−1.12 − 0.364i)20-s + (0.761 + 1.04i)22-s − 1.10·23-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: $\chi_{1512} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.951 + 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3879174743\)
\(L(\frac12)\) \(\approx\) \(0.3879174743\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 - 0.831i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 2.63iT - 5T^{2} \)
11 \( 1 + 4.29iT - 11T^{2} \)
13 \( 1 + 0.0480iT - 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + 0.726iT - 19T^{2} \)
23 \( 1 + 5.28T + 23T^{2} \)
29 \( 1 + 8.00iT - 29T^{2} \)
31 \( 1 - 4.90T + 31T^{2} \)
37 \( 1 - 9.10iT - 37T^{2} \)
41 \( 1 - 1.45T + 41T^{2} \)
43 \( 1 - 8.85iT - 43T^{2} \)
47 \( 1 + 9.23T + 47T^{2} \)
53 \( 1 + 3.14iT - 53T^{2} \)
59 \( 1 - 5.90iT - 59T^{2} \)
61 \( 1 + 2.19iT - 61T^{2} \)
67 \( 1 + 8.55iT - 67T^{2} \)
71 \( 1 + 3.86T + 71T^{2} \)
73 \( 1 + 7.56T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 - 9.05iT - 83T^{2} \)
89 \( 1 - 5.15T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−9.001045410477586385979741558727, −8.271179225752385906772522927963, −7.916942181486237962658587535366, −6.48709337916468786582338459550, −6.10830318775839798533489772129, −5.09174101794811102351191794476, −4.25894564022353755046912883469, −2.75139216639510752849007513000, −1.32824589275056228224526173924, −0.20427221657175972306279684863, 1.82204200064481816322718624228, 2.64679908890243389961512947458, 3.60943708249846670837140689416, 4.53421741890681476100039011103, 6.04925392829316273875702765970, 7.06638073041112046179442138060, 7.19246588923405827387465136837, 8.351638524287497539267874997127, 9.190153936792961358604764070594, 10.01444804054725372937921063983

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