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Properties

Label 2-5292-63.20-c1-0-3
Degree $2$
Conductor $5292$
Sign $-0.842 - 0.539i$
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  + (1.43 + 2.48i)5-s + (−2.34 − 1.35i)11-s + (3.18 − 1.84i)13-s − 6.44·17-s − 3.16i·19-s + (2.59 − 1.49i)23-s + (−1.61 + 2.79i)25-s + (2.48 + 1.43i)29-s + (−8.26 + 4.77i)31-s + 3.41·37-s + (−0.794 − 1.37i)41-s + (−4.67 + 8.10i)43-s + (−5.65 + 9.79i)47-s + 2.49i·53-s − 7.78i·55-s + ⋯
L(s)  = 1  + (0.641 + 1.11i)5-s + (−0.708 − 0.408i)11-s + (0.884 − 0.510i)13-s − 1.56·17-s − 0.725i·19-s + (0.540 − 0.311i)23-s + (−0.322 + 0.558i)25-s + (0.461 + 0.266i)29-s + (−1.48 + 0.857i)31-s + 0.561·37-s + (−0.124 − 0.214i)41-s + (−0.713 + 1.23i)43-s + (−0.824 + 1.42i)47-s + 0.343i·53-s − 1.04i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.842 - 0.539i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (4409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.842 - 0.539i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9953032485\)
\(L(\frac12)\) \(\approx\) \(0.9953032485\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 - 2.48i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.34 + 1.35i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.18 + 1.84i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.44T + 17T^{2} \)
19 \( 1 + 3.16iT - 19T^{2} \)
23 \( 1 + (-2.59 + 1.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.48 - 1.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.26 - 4.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.41T + 37T^{2} \)
41 \( 1 + (0.794 + 1.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.67 - 8.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.65 - 9.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.49iT - 53T^{2} \)
59 \( 1 + (-4.33 - 7.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.566 - 0.327i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.86 + 6.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.86iT - 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + (2.59 - 4.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.92 - 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.29T + 89T^{2} \)
97 \( 1 + (-13.2 - 7.62i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.557358721311884801180980988087, −7.73273386221925336624615770288, −6.79022817430586402421046982104, −6.53270393074477177994463451947, −5.67691671858621647648197923789, −4.94576215682043942531091961604, −3.95954043162004733976901799803, −2.87986289513535484576702588656, −2.61077597148380401578467119704, −1.32434221934411890513969451053, 0.24595013768638145295127481707, 1.64860736397670931499878013463, 2.11968199140672612391959246212, 3.46241276017274136131555956148, 4.32592521202480991486513829362, 5.00276691329104001646494184277, 5.65060329481347850879049005504, 6.39661251384035228661531734015, 7.15713348020583380281462458499, 8.043756597817559701668537359197

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