↑

Properties

Label 2-425-17.2-c1-0-21
Degree $2$
Conductor $425$
Sign $0.981 + 0.189i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
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  + (0.813 + 0.813i)2-s + (1.89 − 0.786i)3-s − 0.677i·4-s + (2.18 + 0.904i)6-s + (0.143 − 0.346i)7-s + (2.17 − 2.17i)8-s + (0.866 − 0.866i)9-s + (0.0511 + 0.0211i)11-s + (−0.532 − 1.28i)12-s + 0.388i·13-s + (0.398 − 0.165i)14-s + 2.18·16-s + (−4.08 − 0.594i)17-s + 1.40·18-s + (1.25 + 1.25i)19-s + ⋯
L(s)  = 1  + (0.575 + 0.575i)2-s + (1.09 − 0.454i)3-s − 0.338i·4-s + (0.891 + 0.369i)6-s + (0.0543 − 0.131i)7-s + (0.769 − 0.769i)8-s + (0.288 − 0.288i)9-s + (0.0154 + 0.00639i)11-s + (−0.153 − 0.371i)12-s + 0.107i·13-s + (0.106 − 0.0441i)14-s + 0.546·16-s + (−0.989 − 0.144i)17-s + 0.332·18-s + (0.287 + 0.287i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.981 + 0.189i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.981 + 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.57171 - 0.246532i\)
\(L(\frac12)\) \(\approx\) \(2.57171 - 0.246532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (4.08 + 0.594i)T \)
good2 \( 1 + (-0.813 - 0.813i)T + 2iT^{2} \)
3 \( 1 + (-1.89 + 0.786i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (-0.143 + 0.346i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.0511 - 0.0211i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 0.388iT - 13T^{2} \)
19 \( 1 + (-1.25 - 1.25i)T + 19iT^{2} \)
23 \( 1 + (-0.948 - 0.392i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.23 - 7.80i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-5.59 + 2.31i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (8.77 - 3.63i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.93 + 7.08i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (4.77 - 4.77i)T - 43iT^{2} \)
47 \( 1 + 1.84iT - 47T^{2} \)
53 \( 1 + (6.43 + 6.43i)T + 53iT^{2} \)
59 \( 1 + (9.61 - 9.61i)T - 59iT^{2} \)
61 \( 1 + (2.22 - 5.36i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 + (1.04 - 0.430i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.838 - 2.02i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (3.64 + 1.50i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (6.67 + 6.67i)T + 83iT^{2} \)
89 \( 1 - 2.61iT - 89T^{2} \)
97 \( 1 + (2.01 + 4.85i)T + (-68.5 + 68.5i)T^{2} \)
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

−11.08677137341586041733196629094, −10.17738173047988900781600477230, −9.132993080197832337846846142379, −8.330028800296717865412940522454, −7.27288774008674148911779260335, −6.63169072313655730863751743001, −5.38991975171489226893034950266, −4.35067557353121339739822789792, −3.04729406460627786292373446954, −1.60991926625598015347227448754, 2.21549020727799240269326217034, 3.09891805997689678098532492543, 4.06118160169683352131669175759, 4.95609822308764486165912723614, 6.52620224770816765602865735906, 7.84712163122586239106768569043, 8.493178924467650782278855873502, 9.334855909157935462848305150201, 10.36303571760696286380053339133, 11.32473298739174924628294415275

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