↑

Properties

Label 2-15e2-25.21-c1-0-4
Degree $2$
Conductor $225$
Sign $0.703 - 0.710i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0762 + 0.234i)2-s + (1.56 + 1.13i)4-s + (2.09 + 0.784i)5-s − 1.24·7-s + (−0.786 + 0.571i)8-s + (−0.343 + 0.431i)10-s + (−0.794 + 2.44i)11-s + (−1.44 − 4.45i)13-s + (0.0950 − 0.292i)14-s + (1.12 + 3.46i)16-s + (4.72 − 3.43i)17-s + (−3.37 + 2.45i)19-s + (2.39 + 3.61i)20-s + (−0.512 − 0.372i)22-s + (−0.496 + 1.52i)23-s + ⋯
L(s)  = 1  + (−0.0539 + 0.165i)2-s + (0.784 + 0.569i)4-s + (0.936 + 0.350i)5-s − 0.471·7-s + (−0.278 + 0.201i)8-s + (−0.108 + 0.136i)10-s + (−0.239 + 0.736i)11-s + (−0.401 − 1.23i)13-s + (0.0254 − 0.0781i)14-s + (0.281 + 0.865i)16-s + (1.14 − 0.832i)17-s + (−0.773 + 0.562i)19-s + (0.534 + 0.808i)20-s + (−0.109 − 0.0794i)22-s + (−0.103 + 0.318i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.703 - 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37131 + 0.572421i\)
\(L(\frac12)\) \(\approx\) \(1.37131 + 0.572421i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.09 - 0.784i)T \)
good2 \( 1 + (0.0762 - 0.234i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + 1.24T + 7T^{2} \)
11 \( 1 + (0.794 - 2.44i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.44 + 4.45i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.72 + 3.43i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.37 - 2.45i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.496 - 1.52i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.60 + 1.89i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-7.43 + 5.40i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.394 + 1.21i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.68 + 8.27i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.88T + 43T^{2} \)
47 \( 1 + (2.59 + 1.88i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (10.7 + 7.83i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.97 - 6.07i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.18 + 3.63i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.19 + 5.95i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-11.2 - 8.15i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.51 - 10.8i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.29 + 6.74i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.29 - 1.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.426 + 1.31i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.0246 + 0.0179i)T + (29.9 + 92.2i)T^{2} \)
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

−12.51142301840184324414033474363, −11.44422889150205090292753273111, −10.19248959264830983117519870137, −9.788336825408260298807827805777, −8.187505487282607481009996608069, −7.31856305024090848658524803413, −6.31107375179781042032270831127, −5.32162954334185692844983715863, −3.33988455923528886915769536355, −2.25225657451800932228711755041, 1.57710672046606996181604229614, 2.98262871354911658529801793235, 4.90768152490461678173931545778, 6.14591306146217785627488869738, 6.67366921411022613306843692975, 8.294756469360754134552424534687, 9.462975542955028981040107671520, 10.14901185821853174548391695859, 11.05500465058466289575168762471, 12.11744555159961883546056080247

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