Properties

Degree $2$
Conductor $271950$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 4·11-s − 12-s − 2·13-s + 16-s + 2·17-s − 18-s − 4·19-s − 4·22-s + 8·23-s + 24-s + 2·26-s − 27-s − 2·29-s − 8·31-s − 32-s − 4·33-s − 2·34-s + 36-s − 37-s + 4·38-s + 2·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s − 0.164·37-s + 0.648·38-s + 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 271950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 271950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(271950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 37\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{271950} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 271950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5343124846\)
\(L(\frac12)\) \(\approx\) \(0.5343124846\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
37 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−12.79361984489095, −12.24482600966029, −11.63988932597715, −11.42950593158591, −10.99366017402346, −10.43243520831328, −9.992008114245762, −9.594947041117643, −9.013440876904138, −8.749821618331604, −8.177958727993941, −7.581280065873659, −6.999046555965275, −6.738627002914650, −6.405059069265290, −5.592599557489671, −5.221998565618456, −4.732565236985666, −4.000556792064310, −3.500054626722098, −2.995882513138411, −2.139918631960369, −1.576748461267224, −1.177740787401393, −0.2410533282319465, 0.2410533282319465, 1.177740787401393, 1.576748461267224, 2.139918631960369, 2.995882513138411, 3.500054626722098, 4.000556792064310, 4.732565236985666, 5.221998565618456, 5.592599557489671, 6.405059069265290, 6.738627002914650, 6.999046555965275, 7.581280065873659, 8.177958727993941, 8.749821618331604, 9.013440876904138, 9.594947041117643, 9.992008114245762, 10.43243520831328, 10.99366017402346, 11.42950593158591, 11.63988932597715, 12.24482600966029, 12.79361984489095

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