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Properties

Label 2-59290-1.1-c1-0-20
Degree $2$
Conductor $59290$
Sign $1$
Analytic cond. $473.433$
Root an. cond. $21.7585$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s + 4-s + 5-s − 2·6-s − 8-s + 9-s − 10-s + 2·12-s − 4·13-s + 2·15-s + 16-s − 18-s − 4·19-s + 20-s − 2·24-s + 25-s + 4·26-s − 4·27-s + 6·29-s − 2·30-s + 10·31-s − 32-s + 36-s + 2·37-s + 4·38-s − 8·39-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s − 1.10·13-s + 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.408·24-s + 1/5·25-s + 0.784·26-s − 0.769·27-s + 1.11·29-s − 0.365·30-s + 1.79·31-s − 0.176·32-s + 1/6·36-s + 0.328·37-s + 0.648·38-s − 1.28·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(59290\)    =    \(2 \cdot 5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(473.433\)
Root analytic conductor: \(21.7585\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 59290,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.058396152\)
\(L(\frac12)\) \(\approx\) \(2.058396152\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.36860709636916, −13.94648802893076, −13.37433798518500, −12.92904739830713, −12.19673884547613, −11.87490739952024, −11.21805525700660, −10.40464050643364, −10.20023849892649, −9.554351622686317, −9.269723493616200, −8.542158371770289, −8.134122218743428, −7.930969917785224, −6.933991641843785, −6.710543143723324, −6.029518574889976, −5.236842616525201, −4.625214730357236, −3.996579434218172, −2.915924379055667, −2.882646359621084, −2.095063551758055, −1.536826606719360, −0.4871366117549406, 0.4871366117549406, 1.536826606719360, 2.095063551758055, 2.882646359621084, 2.915924379055667, 3.996579434218172, 4.625214730357236, 5.236842616525201, 6.029518574889976, 6.710543143723324, 6.933991641843785, 7.930969917785224, 8.134122218743428, 8.542158371770289, 9.269723493616200, 9.554351622686317, 10.20023849892649, 10.40464050643364, 11.21805525700660, 11.87490739952024, 12.19673884547613, 12.92904739830713, 13.37433798518500, 13.94648802893076, 14.36860709636916

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