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Properties

Label 2-48510-1.1-c1-0-22
Degree $2$
Conductor $48510$
Sign $1$
Analytic cond. $387.354$
Root an. cond. $19.6813$
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 + 4-s − 5-s − 8-s + 10-s − 11-s − 2·13-s + 16-s + 6·17-s + 4·19-s − 20-s + 22-s − 6·23-s + 25-s + 2·26-s + 6·29-s − 2·31-s − 32-s − 6·34-s + 2·37-s − 4·38-s + 40-s − 12·41-s + 8·43-s − 44-s + 6·46-s + 12·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.648·38-s + 0.158·40-s − 1.87·41-s + 1.21·43-s − 0.150·44-s + 0.884·46-s + 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48510\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(387.354\)
Root analytic conductor: \(19.6813\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 48510,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.538202425\)
\(L(\frac12)\) \(\approx\) \(1.538202425\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.48043922098981, −14.22181706808793, −13.60488586925758, −12.91800356643125, −12.23397978838819, −11.87886493772368, −11.71571082579290, −10.69750050429346, −10.40666088488368, −9.809620983520656, −9.527971436380055, −8.627206527694882, −8.291917206861451, −7.645793650840738, −7.346855185441143, −6.741094898741033, −5.939634383759571, −5.435164573217950, −4.904693201510822, −3.938219997807173, −3.523309129413080, −2.692674136391187, −2.164521563481758, −1.129539778846146, −0.5652165047339811, 0.5652165047339811, 1.129539778846146, 2.164521563481758, 2.692674136391187, 3.523309129413080, 3.938219997807173, 4.904693201510822, 5.435164573217950, 5.939634383759571, 6.741094898741033, 7.346855185441143, 7.645793650840738, 8.291917206861451, 8.627206527694882, 9.527971436380055, 9.809620983520656, 10.40666088488368, 10.69750050429346, 11.71571082579290, 11.87886493772368, 12.23397978838819, 12.91800356643125, 13.60488586925758, 14.22181706808793, 14.48043922098981

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