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Properties

Label 2-72e2-8.5-c1-0-83
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  − 5.44i·11-s + 1.89·17-s − 8.34i·19-s + 5·25-s − 12.7·41-s + 2.34i·43-s − 7·49-s + 9.24i·59-s + 14.3i·67-s + 13.6·73-s − 18i·83-s − 18·89-s − 19.6·97-s − 20.1i·107-s − 18·113-s + ⋯
L(s)  = 1  − 1.64i·11-s + 0.460·17-s − 1.91i·19-s + 25-s − 1.99·41-s + 0.358i·43-s − 49-s + 1.20i·59-s + 1.75i·67-s + 1.60·73-s − 1.97i·83-s − 1.90·89-s − 1.99·97-s − 1.94i·107-s − 1.69·113-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

\(L(1)\) \(\approx\) \(1.230342630\)
\(L(\frac12)\) \(\approx\) \(1.230342630\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 5.44iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 1.89T + 17T^{2} \)
19 \( 1 + 8.34iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 12.7T + 41T^{2} \)
43 \( 1 - 2.34iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 9.24iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 14.3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 18iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + 19.6T + 97T^{2} \)
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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

−8.148094371516945435824197838222, −7.04730774660934441737212207529, −6.64231199841891017754177999419, −5.68951684403784384432210902326, −5.11920402405585130607817942532, −4.24630274993356000878944246483, −3.17199038828512136087874974095, −2.79005849944718192371728581646, −1.34097892078957790745827216928, −0.33326741063559324937090234427, 1.40455734425236748529902949004, 2.09926111403810818244976995845, 3.27715146866284772597946065651, 3.99202336420370624824970231567, 4.91582110528020345258185656905, 5.42554305455222010010252165466, 6.53387578611953141810270462913, 6.92917862205149731704101679822, 7.936384839119191804862419264188, 8.205777131805295096091299689216

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