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Properties

Label 2-72e2-8.5-c1-0-30
Degree $2$
Conductor $5184$
Sign $0.965 - 0.258i$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
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  − 3.46i·5-s − 3i·11-s + 3.46i·13-s + 3·17-s + 7i·19-s + 3.46·23-s − 6.99·25-s + 6.92i·29-s + 6.92·31-s + 10.3i·37-s − 3·41-s + 5i·43-s + 3.46·47-s − 7·49-s + 13.8i·53-s + ⋯
L(s)  = 1  − 1.54i·5-s − 0.904i·11-s + 0.960i·13-s + 0.727·17-s + 1.60i·19-s + 0.722·23-s − 1.39·25-s + 1.28i·29-s + 1.24·31-s + 1.70i·37-s − 0.468·41-s + 0.762i·43-s + 0.505·47-s − 49-s + 1.90i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.965 - 0.258i$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (2593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ 0.965 - 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.801817929\)
\(L(\frac12)\) \(\approx\) \(1.801817929\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 7iT - 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 - 10.3iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 13.8iT - 53T^{2} \)
59 \( 1 + 9iT - 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 5iT - 67T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−8.216566379501256770310523360657, −7.87214962294971078658712979525, −6.66088056850655419594289079331, −6.04416102699831295205526551652, −5.19037312646676671436744835861, −4.71371825477862558385384338227, −3.81178842245117194613814933111, −3.00894750405957261778479864323, −1.53265288700376245641843406697, −1.06834977283347594720119136424, 0.55255197755816318459666949313, 2.15942681793787415171889032176, 2.79305102560338683598428018205, 3.47879435512423321146890355081, 4.48052089464157977807447056244, 5.31845940535296354202389363180, 6.15054483097042206181872647134, 6.87798450915467039462594021458, 7.35291023479982293502657239801, 7.944817097969424267925946916462

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