↑

Properties

Label 2-3825-425.237-c0-0-1
Degree $2$
Conductor $3825$
Sign $0.612 + 0.790i$
Analytic cond. $1.90892$
Root an. cond. $1.38163$
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  + (0.951 − 0.309i)4-s + (0.707 − 0.707i)5-s + (1.04 − 1.44i)11-s + (−0.221 + 1.39i)13-s + (0.809 − 0.587i)16-s + (0.453 + 0.891i)17-s + (−1.80 − 0.587i)19-s + (0.453 − 0.891i)20-s + (−0.253 − 1.59i)23-s − 1.00i·25-s + (−0.0966 − 0.297i)29-s + (1.16 + 1.59i)41-s + (−1.26 + 1.26i)43-s + (0.550 − 1.69i)44-s + i·49-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)4-s + (0.707 − 0.707i)5-s + (1.04 − 1.44i)11-s + (−0.221 + 1.39i)13-s + (0.809 − 0.587i)16-s + (0.453 + 0.891i)17-s + (−1.80 − 0.587i)19-s + (0.453 − 0.891i)20-s + (−0.253 − 1.59i)23-s − 1.00i·25-s + (−0.0966 − 0.297i)29-s + (1.16 + 1.59i)41-s + (−1.26 + 1.26i)43-s + (0.550 − 1.69i)44-s + i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3825\)    =    \(3^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.612 + 0.790i$
Analytic conductor: \(1.90892\)
Root analytic conductor: \(1.38163\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3825} (3637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3825,\ (\ :0),\ 0.612 + 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.911978344\)
\(L(\frac12)\) \(\approx\) \(1.911978344\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-0.453 - 0.891i)T \)
good2 \( 1 + (-0.951 + 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-1.04 + 1.44i)T + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.221 - 1.39i)T + (-0.951 - 0.309i)T^{2} \)
19 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.253 + 1.59i)T + (-0.951 + 0.309i)T^{2} \)
29 \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (-1.16 - 1.59i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
47 \( 1 + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \)
71 \( 1 + (0.863 - 0.280i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.587 + 0.809i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.587 + 0.809i)T^{2} \)
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

−8.622437816816513427569749650497, −8.012992412067247168254013593956, −6.69689503925259665432233534697, −6.22935288688459380883012508532, −6.05072651593314379282098892630, −4.68693723652641585638376714840, −4.11498977294663121970922920437, −2.83950617769150269718323448300, −1.96634604382805964461331435510, −1.14155210228755438596778607108, 1.68796842745572269129910650005, 2.23552011640445445862400917844, 3.25671659761812917556816527157, 3.95915507587039593325684082233, 5.25756190802951804410137676505, 5.88999332175425051729586359139, 6.70441716864833444899815003750, 7.24781461440468457911748114638, 7.74231928173857480892074267091, 8.805388720752172180390762890245

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