↑

Properties

Label 2-245-245.27-c1-0-10
Degree $2$
Conductor $245$
Sign $0.984 + 0.177i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  + (−0.145 − 0.0510i)2-s + (−2.67 + 1.68i)3-s + (−1.54 − 1.23i)4-s + (1.64 − 1.51i)5-s + (0.476 − 0.108i)6-s + (−1.48 + 2.19i)7-s + (0.326 + 0.520i)8-s + (3.03 − 6.29i)9-s + (−0.316 + 0.137i)10-s + (3.31 − 1.59i)11-s + (6.20 + 0.699i)12-s + (0.303 + 0.106i)13-s + (0.328 − 0.243i)14-s + (−1.83 + 6.82i)15-s + (0.858 + 3.76i)16-s + (6.14 + 0.692i)17-s + ⋯
L(s)  = 1  + (−0.103 − 0.0360i)2-s + (−1.54 + 0.970i)3-s + (−0.772 − 0.616i)4-s + (0.733 − 0.679i)5-s + (0.194 − 0.0443i)6-s + (−0.560 + 0.827i)7-s + (0.115 + 0.183i)8-s + (1.01 − 2.09i)9-s + (−0.100 + 0.0435i)10-s + (1.00 − 0.482i)11-s + (1.79 + 0.201i)12-s + (0.0842 + 0.0294i)13-s + (0.0877 − 0.0651i)14-s + (−0.474 + 1.76i)15-s + (0.214 + 0.940i)16-s + (1.48 + 0.167i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.984 + 0.177i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.984 + 0.177i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681635 - 0.0611246i\)
\(L(\frac12)\) \(\approx\) \(0.681635 - 0.0611246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.64 + 1.51i)T \)
7 \( 1 + (1.48 - 2.19i)T \)
good2 \( 1 + (0.145 + 0.0510i)T + (1.56 + 1.24i)T^{2} \)
3 \( 1 + (2.67 - 1.68i)T + (1.30 - 2.70i)T^{2} \)
11 \( 1 + (-3.31 + 1.59i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.303 - 0.106i)T + (10.1 + 8.10i)T^{2} \)
17 \( 1 + (-6.14 - 0.692i)T + (16.5 + 3.78i)T^{2} \)
19 \( 1 - 3.41T + 19T^{2} \)
23 \( 1 + (0.722 + 6.41i)T + (-22.4 + 5.11i)T^{2} \)
29 \( 1 + (1.05 - 0.842i)T + (6.45 - 28.2i)T^{2} \)
31 \( 1 - 2.78iT - 31T^{2} \)
37 \( 1 + (-0.957 + 8.49i)T + (-36.0 - 8.23i)T^{2} \)
41 \( 1 + (-0.693 - 0.158i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (0.709 + 0.445i)T + (18.6 + 38.7i)T^{2} \)
47 \( 1 + (-0.509 + 1.45i)T + (-36.7 - 29.3i)T^{2} \)
53 \( 1 + (-1.14 - 10.1i)T + (-51.6 + 11.7i)T^{2} \)
59 \( 1 + (-0.629 - 2.75i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-6.74 + 5.38i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (1.19 + 1.19i)T + 67iT^{2} \)
71 \( 1 + (-5.42 + 6.80i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-1.45 - 4.15i)T + (-57.0 + 45.5i)T^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 + (-2.87 - 8.22i)T + (-64.8 + 51.7i)T^{2} \)
89 \( 1 + (-3.78 - 1.82i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-0.682 + 0.682i)T - 97iT^{2} \)
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

−12.16042942400878337212090258482, −10.94495056911970816678401670076, −10.00653311510942725535695883568, −9.473482156548310137308211387634, −8.727486060652594472303265575952, −6.36709109739524885905325827851, −5.70931425102592424930414450099, −5.11577503692690113637388043539, −3.88552080358990208012826285222, −0.922389476306553328023851891878, 1.20374732935499918097619953010, 3.55680472351968764365269159479, 5.12615947995576574489585069084, 6.14612382299879197451020261001, 7.11017721790268949941725948385, 7.68029932416471335954896217992, 9.642739287981012017698295400172, 10.09018980028696765248779631422, 11.45528775983502206628563652580, 12.03914244543136591657174792497

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