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Properties

Label 2-72e2-8.5-c1-0-49
Degree $2$
Conductor $5184$
Sign $0.707 + 0.707i$
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  + 3.87i·5-s − 3.87·7-s + 3i·11-s − 3.87i·13-s − 2i·19-s − 3.87·23-s − 10.0·25-s + 3.87i·29-s − 3.87·31-s − 15.0i·35-s − 7.74i·37-s + 9·41-s − 7i·43-s − 3.87·47-s + 8.00·49-s + ⋯
L(s)  = 1  + 1.73i·5-s − 1.46·7-s + 0.904i·11-s − 1.07i·13-s − 0.458i·19-s − 0.807·23-s − 2.00·25-s + 0.719i·29-s − 0.695·31-s − 2.53i·35-s − 1.27i·37-s + 1.40·41-s − 1.06i·43-s − 0.564·47-s + 1.14·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.707 + 0.707i$
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: \((2,\ 5184,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6506205380\)
\(L(\frac12)\) \(\approx\) \(0.6506205380\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.87iT - 5T^{2} \)
7 \( 1 + 3.87T + 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 + 3.87iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 3.87T + 23T^{2} \)
29 \( 1 - 3.87iT - 29T^{2} \)
31 \( 1 + 3.87T + 31T^{2} \)
37 \( 1 + 7.74iT - 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 + 3.87T + 47T^{2} \)
53 \( 1 - 7.74iT - 53T^{2} \)
59 \( 1 + 3iT - 59T^{2} \)
61 \( 1 + 11.6iT - 61T^{2} \)
67 \( 1 - 11iT - 67T^{2} \)
71 \( 1 + 7.74T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 - 3.87T + 79T^{2} \)
83 \( 1 + 3iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
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

−7.82381603018074893879640448809, −7.23438547506677432886210737581, −6.81070130915585471241034357328, −6.06565067484301835209461531626, −5.49215187953267746303535660750, −4.14573474010062086206016142710, −3.42882407918160190508487138136, −2.83751655588806434616108523381, −2.10795434901207246918686510072, −0.21984827749133720401209097180, 0.816968258340082756882501170372, 1.85830905057666613296032775862, 3.06987678743561230661554204782, 3.96587038302867859220209648448, 4.49228644557331374837173766173, 5.52492556381354424861781055472, 6.06928683220274876173884083390, 6.68297339642807233827949084091, 7.79035245307223089602437696626, 8.386700748487506868408618904512

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