Properties

Degree $2$
Conductor $248430$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 2·11-s − 12-s + 15-s + 16-s − 4·17-s − 18-s − 20-s + 2·22-s + 8·23-s + 24-s + 25-s − 27-s − 30-s + 2·31-s − 32-s + 2·33-s + 4·34-s + 36-s − 8·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.223·20-s + 0.426·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.182·30-s + 0.359·31-s − 0.176·32-s + 0.348·33-s + 0.685·34-s + 1/6·36-s − 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(248430\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{248430} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 248430,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 \)
good11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.21314450947603, −12.77792858331514, −12.27816066036965, −11.91064951901690, −11.26248160310652, −10.92816513046064, −10.75994427144686, −10.12343069699008, −9.553649127103613, −9.207285274269744, −8.550757754697633, −8.289397061103494, −7.664807741272079, −7.196208366352714, −6.651833622573543, −6.514876111273683, −5.623149475294347, −5.216258880566225, −4.698531386558014, −4.219394493603567, −3.407972262002506, −2.948342504708704, −2.373094721480620, −1.548692066640241, −1.109635618190887, 0, 0, 1.109635618190887, 1.548692066640241, 2.373094721480620, 2.948342504708704, 3.407972262002506, 4.219394493603567, 4.698531386558014, 5.216258880566225, 5.623149475294347, 6.514876111273683, 6.651833622573543, 7.196208366352714, 7.664807741272079, 8.289397061103494, 8.550757754697633, 9.207285274269744, 9.553649127103613, 10.12343069699008, 10.75994427144686, 10.92816513046064, 11.26248160310652, 11.91064951901690, 12.27816066036965, 12.77792858331514, 13.21314450947603

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