↑

Properties

Label 2-2925-325.272-c0-0-3
Degree $2$
Conductor $2925$
Sign $0.770 + 0.637i$
Analytic cond. $1.45976$
Root an. cond. $1.20820$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.905 + 1.77i)2-s + (−1.74 + 2.40i)4-s + (−0.233 − 0.972i)5-s + (−3.89 − 0.616i)8-s + (1.51 − 1.29i)10-s + (−1.62 − 0.526i)11-s + (−0.891 − 0.453i)13-s + (−1.50 − 4.63i)16-s + (2.74 + 1.13i)20-s + (−0.531 − 3.35i)22-s + (−0.891 + 0.453i)25-s − 1.99i·26-s + (4.09 − 4.09i)32-s + (0.309 + 3.92i)40-s + (−1.23 + 0.401i)41-s + ⋯
L(s)  = 1  + (0.905 + 1.77i)2-s + (−1.74 + 2.40i)4-s + (−0.233 − 0.972i)5-s + (−3.89 − 0.616i)8-s + (1.51 − 1.29i)10-s + (−1.62 − 0.526i)11-s + (−0.891 − 0.453i)13-s + (−1.50 − 4.63i)16-s + (2.74 + 1.13i)20-s + (−0.531 − 3.35i)22-s + (−0.891 + 0.453i)25-s − 1.99i·26-s + (4.09 − 4.09i)32-s + (0.309 + 3.92i)40-s + (−1.23 + 0.401i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (2872, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.770 + 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2226439225\)
\(L(\frac12)\) \(\approx\) \(0.2226439225\)
\(L(1)\) 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 \( 1 + (0.233 + 0.972i)T \)
13 \( 1 + (0.891 + 0.453i)T \)
good2 \( 1 + (-0.905 - 1.77i)T + (-0.587 + 0.809i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (1.62 + 0.526i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.587 + 0.809i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (1.23 - 0.401i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
47 \( 1 + (1.28 - 0.203i)T + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.600 + 1.84i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.951 - 0.309i)T^{2} \)
71 \( 1 + (-0.614 + 0.845i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.154 - 0.0245i)T + (0.951 + 0.309i)T^{2} \)
89 \( 1 + (0.236 - 0.727i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.951 - 0.309i)T^{2} \)
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

−8.332084192091379749183330671827, −8.003819804149616723943527574850, −7.44091095017712295489552831183, −6.50283435487640379470182577638, −5.59195373013506493920272805443, −5.12945510369679493798898042734, −4.63084232394201052232006572692, −3.57706382207566834066219499331, −2.71161796146616772275271066422, −0.094508845380614241131495974781, 1.85983452655965294572959895420, 2.58934752600722832664246735293, 3.16853950897692591180493397750, 4.17003907704953877744905227573, 4.90958269398491980052431552103, 5.55621202099412222805427415151, 6.54533148470268522708289179825, 7.48315815952309111948370010063, 8.491022002358227266480390559608, 9.511405350825788017124938245361

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