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Properties

Label 2-1053-13.12-c1-0-2
Degree $2$
Conductor $1053$
Sign $0.554 + 0.832i$
Analytic cond. $8.40824$
Root an. cond. $2.89969$
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  + 2.61i·2-s − 4.85·4-s + 2.61i·5-s + 4.85i·7-s − 7.47i·8-s − 6.85·10-s − 3.38i·11-s + (−2 − 3i)13-s − 12.7·14-s + 9.85·16-s − 3·17-s + 6.70i·19-s − 12.7i·20-s + 8.85·22-s − 1.14·23-s + ⋯
L(s)  = 1  + 1.85i·2-s − 2.42·4-s + 1.17i·5-s + 1.83i·7-s − 2.64i·8-s − 2.16·10-s − 1.01i·11-s + (−0.554 − 0.832i)13-s − 3.39·14-s + 2.46·16-s − 0.727·17-s + 1.53i·19-s − 2.84i·20-s + 1.88·22-s − 0.238·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.554 + 0.832i$
Analytic conductor: \(8.40824\)
Root analytic conductor: \(2.89969\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1053,\ (\ :1/2),\ 0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7358834222\)
\(L(\frac12)\) \(\approx\) \(0.7358834222\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (2 + 3i)T \)
good2 \( 1 - 2.61iT - 2T^{2} \)
5 \( 1 - 2.61iT - 5T^{2} \)
7 \( 1 - 4.85iT - 7T^{2} \)
11 \( 1 + 3.38iT - 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 6.70iT - 19T^{2} \)
23 \( 1 + 1.14T + 23T^{2} \)
29 \( 1 + 1.85T + 29T^{2} \)
31 \( 1 - 1.85iT - 31T^{2} \)
37 \( 1 - 1.14iT - 37T^{2} \)
41 \( 1 + 1.09iT - 41T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 - 1.47iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 + 2.85T + 61T^{2} \)
67 \( 1 - 4.14iT - 67T^{2} \)
71 \( 1 - 3.76iT - 71T^{2} \)
73 \( 1 - 6.70iT - 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 - 1.47iT - 83T^{2} \)
89 \( 1 + 6.38iT - 89T^{2} \)
97 \( 1 + 9.70iT - 97T^{2} \)
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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

−10.32060476154012964946934800709, −9.425847004904816243490321494239, −8.605670854703892321756188512918, −8.083846759164850757599275131895, −7.20264008361883384888800778776, −6.17693665638201299434087223107, −5.88032173625355434476014638914, −5.10538344993652355935536789997, −3.64509898056970800350878528431, −2.56590474484180694495779948606, 0.35054361063876387162813206069, 1.38703611336561299415088507410, 2.47089694927021293220025756260, 4.02116971063467184599825523445, 4.41185230553677473591920755164, 4.99208286788931353351411261474, 6.88266494470479195649624525152, 7.71246386463547796851859543210, 9.001272674711060797336676656423, 9.323763006511920532043609210899

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