Properties

Degree $2$
Conductor $57498$
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 − 3-s + 4-s + 2·5-s + 6-s − 7-s − 8-s + 9-s − 2·10-s − 4·11-s − 12-s − 6·13-s + 14-s − 2·15-s + 16-s − 2·17-s − 18-s + 4·19-s + 2·20-s + 21-s + 4·22-s − 8·23-s + 24-s − 25-s + 6·26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s − 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57498\)    =    \(2 \cdot 3 \cdot 7 \cdot 37^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{57498} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 57498,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
37 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−14.64122237504473, −13.92433733254927, −13.71809604555050, −13.01400336703566, −12.43920256788808, −12.10118858525421, −11.57931901331636, −10.88354391399767, −10.27440147924864, −10.03616574640424, −9.641015355086366, −9.176140614799169, −8.308082593510263, −7.809132616016500, −7.320568134055959, −6.806445301454516, −6.082907373405239, −5.717769197377252, −5.090500520263713, −4.647781074565769, −3.673633074964561, −2.840671428801989, −2.227705012320293, −1.880046146227356, −0.6829364616712145, 0, 0.6829364616712145, 1.880046146227356, 2.227705012320293, 2.840671428801989, 3.673633074964561, 4.647781074565769, 5.090500520263713, 5.717769197377252, 6.082907373405239, 6.806445301454516, 7.320568134055959, 7.809132616016500, 8.308082593510263, 9.176140614799169, 9.641015355086366, 10.03616574640424, 10.27440147924864, 10.88354391399767, 11.57931901331636, 12.10118858525421, 12.43920256788808, 13.01400336703566, 13.71809604555050, 13.92433733254927, 14.64122237504473

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