Properties

Degree $2$
Conductor $1575$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·4-s + 7·7-s + 24·8-s + 21·11-s − 24·13-s − 14·14-s − 16·16-s − 22·17-s + 16·19-s − 42·22-s − 25·23-s + 48·26-s − 28·28-s − 167·29-s + 10·31-s − 160·32-s + 44·34-s + 133·37-s − 32·38-s + 168·41-s + 97·43-s − 84·44-s + 50·46-s − 400·47-s + 49·49-s + 96·52-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s + 0.575·11-s − 0.512·13-s − 0.267·14-s − 1/4·16-s − 0.313·17-s + 0.193·19-s − 0.407·22-s − 0.226·23-s + 0.362·26-s − 0.188·28-s − 1.06·29-s + 0.0579·31-s − 0.883·32-s + 0.221·34-s + 0.590·37-s − 0.136·38-s + 0.639·41-s + 0.344·43-s − 0.287·44-s + 0.160·46-s − 1.24·47-s + 1/7·49-s + 0.256·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(3\)
Character: $\chi_{1575} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1575,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 - p T \)
good2 \( 1 + p T + p^{3} T^{2} \)
11 \( 1 - 21 T + p^{3} T^{2} \)
13 \( 1 + 24 T + p^{3} T^{2} \)
17 \( 1 + 22 T + p^{3} T^{2} \)
19 \( 1 - 16 T + p^{3} T^{2} \)
23 \( 1 + 25 T + p^{3} T^{2} \)
29 \( 1 + 167 T + p^{3} T^{2} \)
31 \( 1 - 10 T + p^{3} T^{2} \)
37 \( 1 - 133 T + p^{3} T^{2} \)
41 \( 1 - 168 T + p^{3} T^{2} \)
43 \( 1 - 97 T + p^{3} T^{2} \)
47 \( 1 + 400 T + p^{3} T^{2} \)
53 \( 1 + 182 T + p^{3} T^{2} \)
59 \( 1 + 488 T + p^{3} T^{2} \)
61 \( 1 - 28 T + p^{3} T^{2} \)
67 \( 1 - 967 T + p^{3} T^{2} \)
71 \( 1 - 285 T + p^{3} T^{2} \)
73 \( 1 - 838 T + p^{3} T^{2} \)
79 \( 1 + 469 T + p^{3} T^{2} \)
83 \( 1 + 406 T + p^{3} T^{2} \)
89 \( 1 + 324 T + p^{3} T^{2} \)
97 \( 1 - 114 T + p^{3} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.738834470075254363302823170890, −7.943612717346804702154437886914, −7.34510187854851781794222580692, −6.33001330185905299587651802807, −5.25050266951303755039743593152, −4.48874392528107057334571383838, −3.62119398716569122686394815600, −2.18074656671858377977190330205, −1.14973074640362935848954700088, 0, 1.14973074640362935848954700088, 2.18074656671858377977190330205, 3.62119398716569122686394815600, 4.48874392528107057334571383838, 5.25050266951303755039743593152, 6.33001330185905299587651802807, 7.34510187854851781794222580692, 7.943612717346804702154437886914, 8.738834470075254363302823170890

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