Properties

Degree $2$
Conductor $59150$
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 + 4-s − 7-s + 8-s − 3·9-s − 4·11-s − 14-s + 16-s − 2·17-s − 3·18-s − 4·22-s − 28-s + 6·29-s − 8·31-s + 32-s − 2·34-s − 3·36-s − 10·37-s − 2·41-s − 4·43-s − 4·44-s + 8·47-s + 49-s + 2·53-s − 56-s + 6·58-s + 8·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s − 1.20·11-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.852·22-s − 0.188·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.133·56-s + 0.787·58-s + 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(59150\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{59150} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 59150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.171189890\)
\(L(\frac12)\) \(\approx\) \(1.171189890\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
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

−14.20095687594553, −13.71217946323455, −13.48300649915085, −12.85228885562701, −12.17351807879080, −12.11642549393581, −11.23064794454889, −10.78290681777153, −10.47446249824106, −9.825057693388684, −9.053365148732477, −8.655559310173651, −8.055473528558796, −7.501914796856152, −6.842637626106368, −6.432219058261266, −5.579124051254289, −5.408803327247201, −4.799926476431122, −4.012886902871024, −3.401267441007611, −2.799427949849874, −2.366835609923972, −1.528529810616742, −0.3071963739223118, 0.3071963739223118, 1.528529810616742, 2.366835609923972, 2.799427949849874, 3.401267441007611, 4.012886902871024, 4.799926476431122, 5.408803327247201, 5.579124051254289, 6.432219058261266, 6.842637626106368, 7.501914796856152, 8.055473528558796, 8.655559310173651, 9.053365148732477, 9.825057693388684, 10.47446249824106, 10.78290681777153, 11.23064794454889, 12.11642549393581, 12.17351807879080, 12.85228885562701, 13.48300649915085, 13.71217946323455, 14.20095687594553

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