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Properties

Label 2-5292-9.4-c1-0-35
Degree $2$
Conductor $5292$
Sign $-0.999 + 0.0167i$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
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  + (−0.119 + 0.207i)5-s + (−2.56 − 4.43i)11-s + (−2.44 + 4.23i)13-s + 3.70·17-s − 3.66·19-s + (3.71 − 6.42i)23-s + (2.47 + 4.28i)25-s + (1.73 + 3.00i)29-s + (−0.358 + 0.621i)31-s + 4.60·37-s + (−2.80 + 4.85i)41-s + (−6.24 − 10.8i)43-s + (−2.16 − 3.75i)47-s + 0.942·53-s + 1.22·55-s + ⋯
L(s)  = 1  + (−0.0534 + 0.0926i)5-s + (−0.772 − 1.33i)11-s + (−0.677 + 1.17i)13-s + 0.898·17-s − 0.839·19-s + (0.773 − 1.34i)23-s + (0.494 + 0.856i)25-s + (0.321 + 0.557i)29-s + (−0.0644 + 0.111i)31-s + 0.756·37-s + (−0.437 + 0.757i)41-s + (−0.952 − 1.64i)43-s + (−0.316 − 0.548i)47-s + 0.129·53-s + 0.165·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.999 + 0.0167i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (1765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.999 + 0.0167i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08353787865\)
\(L(\frac12)\) \(\approx\) \(0.08353787865\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.119 - 0.207i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.56 + 4.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.44 - 4.23i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.70T + 17T^{2} \)
19 \( 1 + 3.66T + 19T^{2} \)
23 \( 1 + (-3.71 + 6.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.73 - 3.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.358 - 0.621i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.60T + 37T^{2} \)
41 \( 1 + (2.80 - 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.24 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.16 + 3.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.942T + 53T^{2} \)
59 \( 1 + (3.78 - 6.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.75 - 4.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.330 + 0.571i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 - 3.66T + 73T^{2} \)
79 \( 1 + (-3.11 - 5.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.85 + 8.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.48T + 89T^{2} \)
97 \( 1 + (8.57 + 14.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−7.88591171470885897374861064683, −7.02005209428182280288650899502, −6.53248689973813685254001920535, −5.58843979818599465375968231763, −4.98088949098891623924285967739, −4.13391100914621975655727657490, −3.15893539557041639419545038536, −2.53528798175306244746658304863, −1.31106650498533566886283660900, −0.02232504985609695210910447064, 1.35466814228488840296802861292, 2.48847934223336012876421267245, 3.10999637586655467719337089524, 4.24562268104698523106412968251, 4.97970737563236898610486724985, 5.48049129992527816838914618889, 6.45817752599606921060695150390, 7.23848278358833136992955481368, 7.913220493172385559308198531290, 8.230821513987130616177004146176

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