Properties

Degree $2$
Conductor $67270$
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 − 7-s + 8-s − 3·9-s − 10-s − 4·11-s + 6·13-s − 14-s + 16-s − 2·17-s − 3·18-s − 20-s − 4·22-s + 25-s + 6·26-s − 28-s − 6·29-s + 32-s − 2·34-s + 35-s − 3·36-s + 10·37-s − 40-s + 2·41-s − 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 9-s − 0.316·10-s − 1.20·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.223·20-s − 0.852·22-s + 1/5·25-s + 1.17·26-s − 0.188·28-s − 1.11·29-s + 0.176·32-s − 0.342·34-s + 0.169·35-s − 1/2·36-s + 1.64·37-s − 0.158·40-s + 0.312·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(67270\)    =    \(2 \cdot 5 \cdot 7 \cdot 31^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{67270} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 67270,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
31 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.40557551282193, −13.75817640186099, −13.46717079808303, −12.91977254857580, −12.70737938178633, −11.80404203936493, −11.38977117278402, −11.11111986488491, −10.54713243737219, −10.08248182333917, −9.161815168205022, −8.790394399477969, −8.239955920424840, −7.686447908656617, −7.232805024414174, −6.420732571323069, −5.873922752766448, −5.706361446600760, −4.909744355842768, −4.229462945361590, −3.736879162745921, −3.057672444931041, −2.673790557247415, −1.879276786244882, −0.8746553641292509, 0, 0.8746553641292509, 1.879276786244882, 2.673790557247415, 3.057672444931041, 3.736879162745921, 4.229462945361590, 4.909744355842768, 5.706361446600760, 5.873922752766448, 6.420732571323069, 7.232805024414174, 7.686447908656617, 8.239955920424840, 8.790394399477969, 9.161815168205022, 10.08248182333917, 10.54713243737219, 11.11111986488491, 11.38977117278402, 11.80404203936493, 12.70737938178633, 12.91977254857580, 13.46717079808303, 13.75817640186099, 14.40557551282193

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