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Properties

Label 2-2184-13.3-c1-0-4
Degree $2$
Conductor $2184$
Sign $-0.999 - 0.0377i$
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 − 1.09·5-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.778 − 1.34i)11-s + (0.778 + 3.52i)13-s + (−0.545 − 0.944i)15-s + (−1.17 + 2.03i)17-s + (1.59 − 2.75i)19-s − 0.999·21-s + (2.21 + 3.84i)23-s − 3.81·25-s − 0.999·27-s + (−0.815 − 1.41i)29-s + 1.24·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 0.487·5-s + (−0.188 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.234 − 0.406i)11-s + (0.215 + 0.976i)13-s + (−0.140 − 0.243i)15-s + (−0.284 + 0.492i)17-s + (0.364 − 0.631i)19-s − 0.218·21-s + (0.462 + 0.800i)23-s − 0.762·25-s − 0.192·27-s + (−0.151 − 0.262i)29-s + 0.223·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2184\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.999 - 0.0377i$
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: \((2,\ 2184,\ (\ :1/2),\ -0.999 - 0.0377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6056733941\)
\(L(\frac12)\) \(\approx\) \(0.6056733941\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.778 - 3.52i)T \)
good5 \( 1 + 1.09T + 5T^{2} \)
11 \( 1 + (0.778 + 1.34i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.17 - 2.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.59 + 2.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.21 - 3.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.815 + 1.41i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.24T + 31T^{2} \)
37 \( 1 + (-0.265 - 0.460i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.66 + 8.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.02 - 6.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + 2.26T + 53T^{2} \)
59 \( 1 + (4.74 - 8.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.220 + 0.381i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.861 - 1.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.60 + 2.78i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.77T + 73T^{2} \)
79 \( 1 + 1.02T + 79T^{2} \)
83 \( 1 + 15.9T + 83T^{2} \)
89 \( 1 + (4.55 + 7.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.68 - 9.85i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.401043110730273119020208271985, −8.699931110880617002250352025336, −8.053763228121841680010033156171, −7.16417738153290183568621492883, −6.32284641200114812944458760183, −5.41166902141366279724869704411, −4.52879941294789711888294641250, −3.70865638767952590446113975222, −2.90844561504404495268304480862, −1.68221532637927088841683081895, 0.19713376491269537682803568601, 1.56389406279143282858203305391, 2.84457505138672753016934343911, 3.56216431913772930924061002282, 4.62653949436079453917860323853, 5.50687203958494383746846164802, 6.53394524720418772598508932952, 7.15868107074996841624976189998, 8.051927127026116102668709929605, 8.331095085081554124708666790681

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