Properties

Label 2-3450-1.1-c1-0-0
Degree $2$
Conductor $3450$
Sign $1$
Analytic cond. $27.5483$
Root an. cond. $5.24865$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 2·11-s − 12-s − 4·13-s + 16-s − 6·17-s − 18-s − 8·19-s + 2·22-s + 23-s + 24-s + 4·26-s − 27-s + 4·29-s − 32-s + 2·33-s + 6·34-s + 36-s + 2·37-s + 8·38-s + 4·39-s − 2·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.83·19-s + 0.426·22-s + 0.208·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.742·29-s − 0.176·32-s + 0.348·33-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 1.29·38-s + 0.640·39-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(27.5483\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5394908177\)
\(L(\frac12)\) \(\approx\) \(0.5394908177\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
23 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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

−8.591556920910459989337540257739, −7.951083160282523591332931975508, −6.94342275656798629514371947037, −6.64557875219879895552398018651, −5.66654127407213124932797278900, −4.79124720691970822537754330640, −4.10569325074294583770187706757, −2.62297255271741741004507829590, −2.04508758036819954688449020522, −0.46933715668305309924262465445, 0.46933715668305309924262465445, 2.04508758036819954688449020522, 2.62297255271741741004507829590, 4.10569325074294583770187706757, 4.79124720691970822537754330640, 5.66654127407213124932797278900, 6.64557875219879895552398018651, 6.94342275656798629514371947037, 7.951083160282523591332931975508, 8.591556920910459989337540257739

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