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Properties

Label 2-2184-13.3-c1-0-35
Degree $2$
Conductor $2184$
Sign $0.231 + 0.972i$
Analytic cond. $17.4393$
Root an. cond. $4.17604$
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)3-s + 1.33·5-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−3.18 − 5.51i)11-s + (3.18 − 1.69i)13-s + (0.669 + 1.15i)15-s + (2.75 − 4.76i)17-s + (−0.839 + 1.45i)19-s − 0.999·21-s + (−2.91 − 5.05i)23-s − 3.20·25-s − 0.999·27-s + (−2.94 − 5.09i)29-s − 9.63·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 0.598·5-s + (−0.188 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.959 − 1.66i)11-s + (0.882 − 0.469i)13-s + (0.172 + 0.299i)15-s + (0.666 − 1.15i)17-s + (−0.192 + 0.333i)19-s − 0.218·21-s + (−0.608 − 1.05i)23-s − 0.641·25-s − 0.192·27-s + (−0.546 − 0.946i)29-s − 1.73·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2184\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.231 + 0.972i$
Analytic conductor: \(17.4393\)
Root analytic conductor: \(4.17604\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2184} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2184,\ (\ :1/2),\ 0.231 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.523616673\)
\(L(\frac12)\) \(\approx\) \(1.523616673\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-3.18 + 1.69i)T \)
good5 \( 1 - 1.33T + 5T^{2} \)
11 \( 1 + (3.18 + 5.51i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.75 + 4.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.839 - 1.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.91 + 5.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.94 + 5.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.63T + 31T^{2} \)
37 \( 1 + (-0.875 - 1.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.98 - 3.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.71 + 2.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.51T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + (0.831 - 1.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.04 + 3.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.45 + 4.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.93 + 13.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + (-6.92 - 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.77 + 9.99i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.974992152403358842279284301116, −8.153948009436503289612566190869, −7.64995052116478333903849184630, −6.15014547665282674426801561347, −5.82140068050804801879596389584, −5.07415507726715238058164184109, −3.77596963490862204270786596741, −3.08436862250156462418783061702, −2.16971089013726357915042173978, −0.47899502975663180084160069996, 1.58686802885203501601776512154, 2.10478038718517297671998710161, 3.46765147037761837456980989068, 4.25769863088344881555871353683, 5.49244316149025052600434123552, 6.00229407971541657670354851425, 7.17045382441671391044698858350, 7.46777785470683028357777800467, 8.408312317655394398572690866810, 9.325437748065897124557699909174

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