↑

Properties

Label 2-3225-129.59-c0-0-0
Degree $2$
Conductor $3225$
Sign $0.914 + 0.403i$
Analytic cond. $1.60948$
Root an. cond. $1.26865$
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  + (0.222 − 0.974i)3-s + (−0.900 + 0.433i)4-s − 1.24·7-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)12-s + (1.12 + 1.40i)13-s + (0.623 − 0.781i)16-s + (0.400 − 0.193i)19-s + (−0.277 + 1.21i)21-s + (−0.623 + 0.781i)27-s + (1.12 − 0.541i)28-s + (0.0990 + 0.433i)31-s + 36-s + 1.80·37-s + (1.62 − 0.781i)39-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)3-s + (−0.900 + 0.433i)4-s − 1.24·7-s + (−0.900 − 0.433i)9-s + (0.222 + 0.974i)12-s + (1.12 + 1.40i)13-s + (0.623 − 0.781i)16-s + (0.400 − 0.193i)19-s + (−0.277 + 1.21i)21-s + (−0.623 + 0.781i)27-s + (1.12 − 0.541i)28-s + (0.0990 + 0.433i)31-s + 36-s + 1.80·37-s + (1.62 − 0.781i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3225\)    =    \(3 \cdot 5^{2} \cdot 43\)
Sign: $0.914 + 0.403i$
Analytic conductor: \(1.60948\)
Root analytic conductor: \(1.26865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3225} (3026, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3225,\ (\ :0),\ 0.914 + 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9053120670\)
\(L(\frac12)\) \(\approx\) \(0.9053120670\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 \)
43 \( 1 + (-0.222 + 0.974i)T \)
good2 \( 1 + (0.900 - 0.433i)T^{2} \)
7 \( 1 + 1.24T + T^{2} \)
11 \( 1 + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
37 \( 1 - 1.80T + T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 + 0.974i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
67 \( 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.900 + 0.433i)T^{2} \)
97 \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \)
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

−8.837182170565213389351402977042, −8.099166438152799104538480605264, −7.29703693013220262250756251501, −6.52058186535694443598923590982, −6.06696780825137633454723189026, −4.94530793408742738344993583513, −3.82592783669284343884907913671, −3.35384367227421140972364526251, −2.24698654105002325576613359272, −0.865650585768402433294956291119, 0.818607339667592748904451010674, 2.74919994475646376138172902654, 3.48818505366698412657961549022, 4.10216095253257874934054648197, 5.05592604498185798140843248234, 5.89241721178063080369769617202, 6.19103021202321077111008912521, 7.66841427353778496452218617820, 8.354877474239826737967514530602, 9.013715579916089690023200225717

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