Properties

Degree $2$
Conductor $349830$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 8-s + 10-s + 4·11-s + 16-s + 6·17-s − 4·19-s + 20-s + 4·22-s + 23-s + 25-s + 2·29-s + 32-s + 6·34-s + 2·37-s − 4·38-s + 40-s + 10·41-s − 4·43-s + 4·44-s + 46-s − 7·49-s + 50-s − 6·53-s + 4·55-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1.20·11-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.371·29-s + 0.176·32-s + 1.02·34-s + 0.328·37-s − 0.648·38-s + 0.158·40-s + 1.56·41-s − 0.609·43-s + 0.603·44-s + 0.147·46-s − 49-s + 0.141·50-s − 0.824·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349830\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{349830} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 349830,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 \)
23 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.71424268965168, −12.49994302205694, −11.88152942892547, −11.49088449800047, −11.13527905555136, −10.44193423102630, −10.18424016351013, −9.596004038502540, −9.200738663888432, −8.709248277715157, −8.150417630697633, −7.574503774257557, −7.261547161326523, −6.503830471755341, −6.144924785107276, −5.999873487192241, −5.179177879618700, −4.789796847952658, −4.257002556630372, −3.732811916655277, −3.242983630828381, −2.714078570850892, −2.089385534716637, −1.333976646116673, −1.138103075642842, 0, 1.138103075642842, 1.333976646116673, 2.089385534716637, 2.714078570850892, 3.242983630828381, 3.732811916655277, 4.257002556630372, 4.789796847952658, 5.179177879618700, 5.999873487192241, 6.144924785107276, 6.503830471755341, 7.261547161326523, 7.574503774257557, 8.150417630697633, 8.709248277715157, 9.200738663888432, 9.596004038502540, 10.18424016351013, 10.44193423102630, 11.13527905555136, 11.49088449800047, 11.88152942892547, 12.49994302205694, 12.71424268965168

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