↑

Properties

Label 2-245-245.223-c1-0-18
Degree $2$
Conductor $245$
Sign $0.971 - 0.235i$
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  + (−0.879 + 2.51i)2-s + (0.183 + 0.291i)3-s + (−3.97 − 3.17i)4-s + (0.772 − 2.09i)5-s + (−0.894 + 0.204i)6-s + (0.182 − 2.63i)7-s + (6.96 − 4.37i)8-s + (1.25 − 2.59i)9-s + (4.59 + 3.78i)10-s + (−4.64 + 2.23i)11-s + (0.196 − 1.74i)12-s + (1.82 − 5.22i)13-s + (6.47 + 2.78i)14-s + (0.753 − 0.159i)15-s + (2.60 + 11.4i)16-s + (−0.0741 + 0.658i)17-s + ⋯
L(s)  = 1  + (−0.621 + 1.77i)2-s + (0.105 + 0.168i)3-s + (−1.98 − 1.58i)4-s + (0.345 − 0.938i)5-s + (−0.365 + 0.0833i)6-s + (0.0690 − 0.997i)7-s + (2.46 − 1.54i)8-s + (0.416 − 0.865i)9-s + (1.45 + 1.19i)10-s + (−1.39 + 0.674i)11-s + (0.0566 − 0.502i)12-s + (0.507 − 1.44i)13-s + (1.72 + 0.743i)14-s + (0.194 − 0.0411i)15-s + (0.651 + 2.85i)16-s + (−0.0179 + 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.971 - 0.235i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.971 - 0.235i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759192 + 0.0907259i\)
\(L(\frac12)\) \(\approx\) \(0.759192 + 0.0907259i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.772 + 2.09i)T \)
7 \( 1 + (-0.182 + 2.63i)T \)
good2 \( 1 + (0.879 - 2.51i)T + (-1.56 - 1.24i)T^{2} \)
3 \( 1 + (-0.183 - 0.291i)T + (-1.30 + 2.70i)T^{2} \)
11 \( 1 + (4.64 - 2.23i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-1.82 + 5.22i)T + (-10.1 - 8.10i)T^{2} \)
17 \( 1 + (0.0741 - 0.658i)T + (-16.5 - 3.78i)T^{2} \)
19 \( 1 + 0.915T + 19T^{2} \)
23 \( 1 + (3.14 - 0.354i)T + (22.4 - 5.11i)T^{2} \)
29 \( 1 + (1.80 - 1.44i)T + (6.45 - 28.2i)T^{2} \)
31 \( 1 - 3.94iT - 31T^{2} \)
37 \( 1 + (-4.62 - 0.521i)T + (36.0 + 8.23i)T^{2} \)
41 \( 1 + (-8.84 - 2.01i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (-2.07 + 3.29i)T + (-18.6 - 38.7i)T^{2} \)
47 \( 1 + (-5.03 - 1.76i)T + (36.7 + 29.3i)T^{2} \)
53 \( 1 + (-11.0 + 1.24i)T + (51.6 - 11.7i)T^{2} \)
59 \( 1 + (0.0306 + 0.134i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (0.121 - 0.0968i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (-7.40 + 7.40i)T - 67iT^{2} \)
71 \( 1 + (-1.53 + 1.92i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (13.6 - 4.77i)T + (57.0 - 45.5i)T^{2} \)
79 \( 1 + 1.46iT - 79T^{2} \)
83 \( 1 + (-2.72 + 0.952i)T + (64.8 - 51.7i)T^{2} \)
89 \( 1 + (8.54 + 4.11i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-7.71 - 7.71i)T + 97iT^{2} \)
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

−12.74103856565226514820884033982, −10.44613729415020070448219295773, −10.00803487673548574232151191829, −8.966303398453489317170452208245, −8.000489791449382408270264152835, −7.40137076229067229453247336195, −6.10208470038001734013734587813, −5.20693254103864434922317076943, −4.16504996389556059684404138700, −0.77171429134882341624051310223, 2.09717286641432844617627693892, 2.66007215099367894549351123104, 4.19980942225402037444363666351, 5.75529395985665317890138666112, 7.54658301401502872612983077775, 8.482366648507329885396654529958, 9.442666356140463232341251865805, 10.34543779416935720115645637002, 11.09956215511539235947263484039, 11.66493132855633415740904563537

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