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Properties

Label 2-1053-9.4-c1-0-27
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  + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s + (1 − 1.73i)5-s + (2 + 3.46i)7-s + 3·8-s + 1.99·10-s + (2 + 3.46i)11-s + (−0.5 + 0.866i)13-s + (−1.99 + 3.46i)14-s + (0.500 + 0.866i)16-s − 2·17-s + (−0.999 − 1.73i)20-s + (−1.99 + 3.46i)22-s + (0.500 + 0.866i)25-s − 0.999·26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + (0.447 − 0.774i)5-s + (0.755 + 1.30i)7-s + 1.06·8-s + 0.632·10-s + (0.603 + 1.04i)11-s + (−0.138 + 0.240i)13-s + (−0.534 + 0.925i)14-s + (0.125 + 0.216i)16-s − 0.485·17-s + (−0.223 − 0.387i)20-s + (−0.426 + 0.738i)22-s + (0.100 + 0.173i)25-s − 0.196·26-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

\(L(1)\) \(\approx\) \(2.640325126\)
\(L(\frac12)\) \(\approx\) \(2.640325126\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5 + 8.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6 - 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.633745672466360067849855892298, −9.306299668418684779114065405649, −8.310363556768826410556548301880, −7.41308088535143123918688974468, −6.44164244713461364188777705659, −5.64905647954319612401538892337, −4.99314562034223084132888497455, −4.27943102738344771103333950024, −2.24582653525589844724095479856, −1.59012569868837490867952876654, 1.25815605945509077861726263599, 2.49081518787413313927800621899, 3.56515168953022369676658345225, 4.18505785172181107275423837379, 5.40127572286425755046353866881, 6.64645270394953870293953988871, 7.23202905044942861984248964359, 8.037389731776452719936335219077, 9.019778270195146997918394372288, 10.23636168968587638932593031171

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