Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.90·2-s − 3-s + 1.62·4-s + 1.90·6-s − 7-s + 0.719·8-s + 9-s + 2·11-s − 1.62·12-s − 6.42·13-s + 1.90·14-s − 4.61·16-s + 4.42·17-s − 1.90·18-s − 2.42·19-s + 21-s − 3.80·22-s + 1.37·23-s − 0.719·24-s + 12.2·26-s − 27-s − 1.62·28-s + 0.755·29-s + 5.18·31-s + 7.34·32-s − 2·33-s − 8.42·34-s + ⋯
L(s)  = 1  − 1.34·2-s − 0.577·3-s + 0.811·4-s + 0.776·6-s − 0.377·7-s + 0.254·8-s + 0.333·9-s + 0.603·11-s − 0.468·12-s − 1.78·13-s + 0.508·14-s − 1.15·16-s + 1.07·17-s − 0.448·18-s − 0.557·19-s + 0.218·21-s − 0.811·22-s + 0.287·23-s − 0.146·24-s + 2.39·26-s − 0.192·27-s − 0.306·28-s + 0.140·29-s + 0.931·31-s + 1.29·32-s − 0.348·33-s − 1.44·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{525} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.498769\)
\(L(\frac12)\)  \(\approx\)  \(0.498769\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
good2 \( 1 + 1.90T + 2T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 6.42T + 13T^{2} \)
17 \( 1 - 4.42T + 17T^{2} \)
19 \( 1 + 2.42T + 19T^{2} \)
23 \( 1 - 1.37T + 23T^{2} \)
29 \( 1 - 0.755T + 29T^{2} \)
31 \( 1 - 5.18T + 31T^{2} \)
37 \( 1 - 7.61T + 37T^{2} \)
41 \( 1 + 8.23T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 - 2.75T + 47T^{2} \)
53 \( 1 - 9.18T + 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 - 6.85T + 61T^{2} \)
67 \( 1 - 2.75T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 1.57T + 73T^{2} \)
79 \( 1 + 4.85T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 4.62T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.48869029531367571657057514899, −9.916486343551225113959428501711, −9.316520234174628371140068914352, −8.230490685348470661746481354447, −7.33870709268040940690541670912, −6.64213211134235603508882279981, −5.34873866904477615142468144811, −4.22140353570702312656420354378, −2.43619215235081207014809590862, −0.799153297890762279963436329692, 0.799153297890762279963436329692, 2.43619215235081207014809590862, 4.22140353570702312656420354378, 5.34873866904477615142468144811, 6.64213211134235603508882279981, 7.33870709268040940690541670912, 8.230490685348470661746481354447, 9.316520234174628371140068914352, 9.916486343551225113959428501711, 10.48869029531367571657057514899

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