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Properties

Label 2-1053-9.4-c1-0-38
Degree $2$
Conductor $1053$
Sign $-0.766 + 0.642i$
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  + (0.207 + 0.358i)2-s + (0.914 − 1.58i)4-s + (−1.41 + 2.44i)5-s + (−1.41 − 2.44i)7-s + 1.58·8-s − 1.17·10-s + (−1 − 1.73i)11-s + (0.5 − 0.866i)13-s + (0.585 − 1.01i)14-s + (−1.49 − 2.59i)16-s − 7.65·17-s − 2.82·19-s + (2.58 + 4.47i)20-s + (0.414 − 0.717i)22-s + (−2 + 3.46i)23-s + ⋯
L(s)  = 1  + (0.146 + 0.253i)2-s + (0.457 − 0.791i)4-s + (−0.632 + 1.09i)5-s + (−0.534 − 0.925i)7-s + 0.560·8-s − 0.370·10-s + (−0.301 − 0.522i)11-s + (0.138 − 0.240i)13-s + (0.156 − 0.271i)14-s + (−0.374 − 0.649i)16-s − 1.85·17-s − 0.648·19-s + (0.578 + 1.00i)20-s + (0.0883 − 0.152i)22-s + (−0.417 + 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(8.40824\)
Root analytic conductor: \(2.89969\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1053,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5652560629\)
\(L(\frac12)\) \(\approx\) \(0.5652560629\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.207 - 0.358i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.41 - 2.44i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.41 + 2.44i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 7.65T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.585 + 1.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 + (-2.58 + 4.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.828 - 1.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.82 + 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (-3.82 + 6.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.65 + 11.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.41 - 5.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 0.343T + 73T^{2} \)
79 \( 1 + (-5.65 - 9.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.82 - 3.16i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + (1.82 + 3.16i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.873386123924989700304730224456, −8.694938739255276973290529644112, −7.64553940551806410071917198557, −6.76286949829648400343269299873, −6.60129117118731816614613495282, −5.39219762465197733562509482306, −4.19353237577099871765797875559, −3.31424573775062588356958554739, −2.09382885740172134262185321631, −0.21849837017514945989382904470, 1.96418247978228395550318465927, 2.87619352380695748619738671982, 4.26748003410696829813051199102, 4.62734526353551417925264808755, 6.08107236841990545254918199923, 6.87756404174636934388452725605, 7.905309854445416646245107337157, 8.717838261796762116415426154648, 9.021146262731224193889856464911, 10.34139582013274642880756133420

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