↑

Properties

Label 2-425-5.4-c3-0-28
Degree $2$
Conductor $425$
Sign $0.894 - 0.447i$
Analytic cond. $25.0758$
Root an. cond. $5.00757$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.64i·2-s − 5.37i·3-s + 5.31·4-s + 8.81·6-s − 2.20i·7-s + 21.8i·8-s − 1.88·9-s + 18.7·11-s − 28.5i·12-s + 62.9i·13-s + 3.60·14-s + 6.68·16-s + 17i·17-s − 3.09i·18-s + 47.8·19-s + ⋯
L(s)  = 1  + 0.579i·2-s − 1.03i·3-s + 0.663·4-s + 0.599·6-s − 0.118i·7-s + 0.964i·8-s − 0.0698·9-s + 0.514·11-s − 0.686i·12-s + 1.34i·13-s + 0.0689·14-s + 0.104·16-s + 0.242i·17-s − 0.0405i·18-s + 0.577·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(25.0758\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.559547491\)
\(L(\frac12)\) \(\approx\) \(2.559547491\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 - 17iT \)
good2 \( 1 - 1.64iT - 8T^{2} \)
3 \( 1 + 5.37iT - 27T^{2} \)
7 \( 1 + 2.20iT - 343T^{2} \)
11 \( 1 - 18.7T + 1.33e3T^{2} \)
13 \( 1 - 62.9iT - 2.19e3T^{2} \)
19 \( 1 - 47.8T + 6.85e3T^{2} \)
23 \( 1 - 153. iT - 1.21e4T^{2} \)
29 \( 1 - 64.4T + 2.43e4T^{2} \)
31 \( 1 + 40.9T + 2.97e4T^{2} \)
37 \( 1 + 32.6iT - 5.06e4T^{2} \)
41 \( 1 - 159.T + 6.89e4T^{2} \)
43 \( 1 + 111. iT - 7.95e4T^{2} \)
47 \( 1 + 614. iT - 1.03e5T^{2} \)
53 \( 1 - 308. iT - 1.48e5T^{2} \)
59 \( 1 + 267.T + 2.05e5T^{2} \)
61 \( 1 - 521.T + 2.26e5T^{2} \)
67 \( 1 + 118. iT - 3.00e5T^{2} \)
71 \( 1 - 1.14e3T + 3.57e5T^{2} \)
73 \( 1 + 40.7iT - 3.89e5T^{2} \)
79 \( 1 + 374.T + 4.93e5T^{2} \)
83 \( 1 - 826. iT - 5.71e5T^{2} \)
89 \( 1 - 38.9T + 7.04e5T^{2} \)
97 \( 1 - 917. iT - 9.12e5T^{2} \)
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

−11.11068090027857833506363755290, −9.831957060178854743437990949131, −8.767976518990893460656588370662, −7.69226416480662668761954316740, −7.05133342759473854331977968521, −6.45728268484996987567925383445, −5.40808572755531706392772493947, −3.87448514002879101247838540678, −2.23294509972993202212436872072, −1.30641850571917803868454402960, 0.956505233259994352199174324282, 2.62800977266097142010115241093, 3.55910511782281963632233152132, 4.64169377381769631497364917854, 5.82553797364197715966600340412, 6.92256857175479013516725102634, 8.014971967377216728859194228964, 9.242437249779718939475982908094, 10.01963762327388303577499394285, 10.64258210614119032539522327220

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