Properties

Degree $2$
Conductor $281775$
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 − 6-s − 3·8-s + 9-s − 4·11-s + 12-s − 13-s − 16-s + 18-s − 4·19-s − 4·22-s + 3·24-s − 26-s − 27-s + 2·29-s + 8·31-s + 5·32-s + 4·33-s − 36-s − 2·37-s − 4·38-s + 39-s − 2·41-s + 4·43-s + 4·44-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.277·13-s − 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.852·22-s + 0.612·24-s − 0.196·26-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.883·32-s + 0.696·33-s − 1/6·36-s − 0.328·37-s − 0.648·38-s + 0.160·39-s − 0.312·41-s + 0.609·43-s + 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(281775\)    =    \(3 \cdot 5^{2} \cdot 13 \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{281775} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 281775,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
5 \( 1 \)
13 \( 1 + T \)
17 \( 1 \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.90525475870083, −12.68094906675646, −12.06691623740783, −11.80068548259619, −11.22089034683833, −10.64107087773913, −10.24762815765257, −9.916132497699575, −9.338162433219187, −8.748568364254669, −8.278666456880743, −7.953833254305433, −7.269594898559256, −6.719958539950215, −6.180901650064265, −5.861990218869029, −5.238478082586779, −4.787283133804468, −4.565209768235303, −3.927451457029654, −3.288752540619294, −2.748130570902686, −2.272450961559735, −1.409919667509117, −0.5681149720156749, 0, 0.5681149720156749, 1.409919667509117, 2.272450961559735, 2.748130570902686, 3.288752540619294, 3.927451457029654, 4.565209768235303, 4.787283133804468, 5.238478082586779, 5.861990218869029, 6.180901650064265, 6.719958539950215, 7.269594898559256, 7.953833254305433, 8.278666456880743, 8.748568364254669, 9.338162433219187, 9.916132497699575, 10.24762815765257, 10.64107087773913, 11.22089034683833, 11.80068548259619, 12.06691623740783, 12.68094906675646, 12.90525475870083

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