Properties

Degree $2$
Conductor $102850$
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 + 2·3-s + 4-s + 2·6-s − 4·7-s + 8-s + 9-s + 2·12-s + 2·13-s − 4·14-s + 16-s − 17-s + 18-s + 4·19-s − 8·21-s + 2·24-s + 2·26-s − 4·27-s − 4·28-s − 4·31-s + 32-s − 34-s + 36-s + 4·37-s + 4·38-s + 4·39-s − 6·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s − 1.74·21-s + 0.408·24-s + 0.392·26-s − 0.769·27-s − 0.755·28-s − 0.718·31-s + 0.176·32-s − 0.171·34-s + 1/6·36-s + 0.657·37-s + 0.648·38-s + 0.640·39-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102850\)    =    \(2 \cdot 5^{2} \cdot 11^{2} \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{102850} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 102850,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 \)
17 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.91053950931011, −13.53853753024058, −13.04549390152354, −12.79760393172315, −12.17105507521054, −11.60980892804517, −11.12711181365634, −10.46683517153739, −9.944109787508779, −9.469098761510089, −9.078260860081109, −8.553371302459231, −7.989592619669638, −7.297518234583180, −7.040408229530392, −6.336072479249808, −5.816282972618054, −5.438863186668760, −4.533976855122470, −3.844698947648796, −3.591424207823819, −2.923403514414460, −2.663009223037015, −1.880799905700271, −1.032474324561604, 0, 1.032474324561604, 1.880799905700271, 2.663009223037015, 2.923403514414460, 3.591424207823819, 3.844698947648796, 4.533976855122470, 5.438863186668760, 5.816282972618054, 6.336072479249808, 7.040408229530392, 7.297518234583180, 7.989592619669638, 8.553371302459231, 9.078260860081109, 9.469098761510089, 9.944109787508779, 10.46683517153739, 11.12711181365634, 11.60980892804517, 12.17105507521054, 12.79760393172315, 13.04549390152354, 13.53853753024058, 13.91053950931011

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