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Properties

Label 2-1824-12.11-c1-0-31
Degree $2$
Conductor $1824$
Sign $0.736 - 0.676i$
Analytic cond. $14.5647$
Root an. cond. $3.81637$
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  + (1.73 + 0.0731i)3-s − 0.691i·5-s + 2.48i·7-s + (2.98 + 0.253i)9-s + 0.168·11-s + 1.70·13-s + (0.0506 − 1.19i)15-s + 6.00i·17-s − i·19-s + (−0.181 + 4.29i)21-s − 3.77·23-s + 4.52·25-s + (5.15 + 0.657i)27-s + 0.299i·29-s + 4.79i·31-s + ⋯
L(s)  = 1  + (0.999 + 0.0422i)3-s − 0.309i·5-s + 0.937i·7-s + (0.996 + 0.0844i)9-s + 0.0508·11-s + 0.473·13-s + (0.0130 − 0.308i)15-s + 1.45i·17-s − 0.229i·19-s + (−0.0396 + 0.936i)21-s − 0.787·23-s + 0.904·25-s + (0.991 + 0.126i)27-s + 0.0556i·29-s + 0.860i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1824\)    =    \(2^{5} \cdot 3 \cdot 19\)
Sign: $0.736 - 0.676i$
Analytic conductor: \(14.5647\)
Root analytic conductor: \(3.81637\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1824} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1824,\ (\ :1/2),\ 0.736 - 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.593532267\)
\(L(\frac12)\) \(\approx\) \(2.593532267\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 0.0731i)T \)
19 \( 1 + iT \)
good5 \( 1 + 0.691iT - 5T^{2} \)
7 \( 1 - 2.48iT - 7T^{2} \)
11 \( 1 - 0.168T + 11T^{2} \)
13 \( 1 - 1.70T + 13T^{2} \)
17 \( 1 - 6.00iT - 17T^{2} \)
23 \( 1 + 3.77T + 23T^{2} \)
29 \( 1 - 0.299iT - 29T^{2} \)
31 \( 1 - 4.79iT - 31T^{2} \)
37 \( 1 + 0.792T + 37T^{2} \)
41 \( 1 - 0.126iT - 41T^{2} \)
43 \( 1 + 1.34iT - 43T^{2} \)
47 \( 1 - 0.240T + 47T^{2} \)
53 \( 1 + 4.06iT - 53T^{2} \)
59 \( 1 - 9.72T + 59T^{2} \)
61 \( 1 + 2.29T + 61T^{2} \)
67 \( 1 - 6.90iT - 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 - 2.57iT - 79T^{2} \)
83 \( 1 + 5.72T + 83T^{2} \)
89 \( 1 + 4.50iT - 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−9.057537398845693527306412514916, −8.610493822200637389530308694689, −8.115257102670599855630051735138, −7.04416680161199344398466149714, −6.18435870948788175206925250083, −5.27712668367228392433303694569, −4.24357698796654565283102121905, −3.42019451827703070581565152659, −2.39443565934857510918774719044, −1.44800626028499724724241154505, 0.927279638248695711134768152072, 2.26802173182499856379031846891, 3.23419095368837060882994357977, 4.02368644844555997817918463346, 4.85346105985439930548581096776, 6.14914499259458953637986795934, 7.07686509423770120357044590018, 7.52203284113503480324091551074, 8.357129124781411809436913049244, 9.174326244904362357302684426945

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