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Properties

Label 2-1512-21.5-c1-0-11
Degree $2$
Conductor $1512$
Sign $0.808 - 0.589i$
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  + (0.680 − 1.17i)5-s + (2.57 − 0.611i)7-s + (−3.45 + 1.99i)11-s + 5.53i·13-s + (0.611 + 1.05i)17-s + (1.83 + 1.05i)19-s + (3.00 + 1.73i)23-s + (1.57 + 2.72i)25-s + 4.14i·29-s + (−1.79 + 1.03i)31-s + (1.03 − 3.44i)35-s + (−0.0588 + 0.101i)37-s + 1.87·41-s + 9.25·43-s + (1.29 − 2.24i)47-s + ⋯
L(s)  = 1  + (0.304 − 0.527i)5-s + (0.972 − 0.231i)7-s + (−1.04 + 0.601i)11-s + 1.53i·13-s + (0.148 + 0.256i)17-s + (0.420 + 0.242i)19-s + (0.627 + 0.362i)23-s + (0.314 + 0.545i)25-s + 0.770i·29-s + (−0.322 + 0.186i)31-s + (0.174 − 0.583i)35-s + (−0.00966 + 0.0167i)37-s + 0.292·41-s + 1.41·43-s + (0.189 − 0.327i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.808 - 0.589i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.808 - 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.854820427\)
\(L(\frac12)\) \(\approx\) \(1.854820427\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.57 + 0.611i)T \)
good5 \( 1 + (-0.680 + 1.17i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.45 - 1.99i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.53iT - 13T^{2} \)
17 \( 1 + (-0.611 - 1.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.83 - 1.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.00 - 1.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.14iT - 29T^{2} \)
31 \( 1 + (1.79 - 1.03i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0588 - 0.101i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.87T + 41T^{2} \)
43 \( 1 - 9.25T + 43T^{2} \)
47 \( 1 + (-1.29 + 2.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.50 - 2.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.60 + 9.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.26 + 0.728i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.895 - 1.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.9iT - 71T^{2} \)
73 \( 1 + (-8.57 + 4.95i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.69 + 2.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.08T + 83T^{2} \)
89 \( 1 + (-7.53 + 13.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−9.292662089445570492261087627238, −8.981191437222610491652511962954, −7.79534326792946762542093360507, −7.37277755742006594898454763400, −6.28631675571802365356260288548, −5.07809839386047299028336289526, −4.83837382728440257142616503171, −3.64972102246909063641500998298, −2.19856283353731933823110216499, −1.34716156508035214987460393842, 0.804119205230955470950028268094, 2.47604579566647265592094995026, 3.02301168174135477239987968499, 4.45571830243808890217809546190, 5.44257614462734229082222742529, 5.84781185746442450907812940180, 7.12860314785061318512121166489, 7.896617247036587254468488032417, 8.385720995919552849587497429544, 9.423289489360860703577660317902

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