Properties

Label 2-245-245.103-c1-0-18
Degree $2$
Conductor $245$
Sign $0.790 + 0.612i$
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  + (−1.17 + 1.59i)2-s + (0.467 − 0.0883i)3-s + (−0.573 − 1.85i)4-s + (−1.51 − 1.64i)5-s + (−0.409 + 0.850i)6-s + (1.41 − 2.23i)7-s + (−0.104 − 0.0364i)8-s + (−2.58 + 1.01i)9-s + (4.41 − 0.479i)10-s + (2.24 − 5.72i)11-s + (−0.431 − 0.817i)12-s + (−2.37 − 0.267i)13-s + (1.90 + 4.89i)14-s + (−0.852 − 0.634i)15-s + (3.39 − 2.31i)16-s + (−2.24 − 0.0838i)17-s + ⋯
L(s)  = 1  + (−0.833 + 1.12i)2-s + (0.269 − 0.0510i)3-s + (−0.286 − 0.929i)4-s + (−0.677 − 0.735i)5-s + (−0.167 + 0.347i)6-s + (0.534 − 0.845i)7-s + (−0.0367 − 0.0128i)8-s + (−0.860 + 0.337i)9-s + (1.39 − 0.151i)10-s + (0.677 − 1.72i)11-s + (−0.124 − 0.235i)12-s + (−0.657 − 0.0741i)13-s + (0.509 + 1.30i)14-s + (−0.220 − 0.163i)15-s + (0.848 − 0.578i)16-s + (−0.543 − 0.0203i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.790 + 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.574913 - 0.196864i\)
\(L(\frac12)\) \(\approx\) \(0.574913 - 0.196864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.51 + 1.64i)T \)
7 \( 1 + (-1.41 + 2.23i)T \)
good2 \( 1 + (1.17 - 1.59i)T + (-0.589 - 1.91i)T^{2} \)
3 \( 1 + (-0.467 + 0.0883i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (-2.24 + 5.72i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (2.37 + 0.267i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (2.24 + 0.0838i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (-1.77 + 3.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.116 - 3.10i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (1.81 - 0.414i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-6.67 + 3.85i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.98 + 1.57i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (0.626 + 1.30i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-1.96 - 5.60i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (5.99 + 4.42i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (4.60 + 2.43i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (0.849 + 11.3i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (0.381 - 1.23i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (3.17 + 0.851i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.34 - 14.6i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-1.88 + 1.39i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (-3.91 - 2.26i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.66 + 14.7i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (-4.43 - 11.3i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-6.01 - 6.01i)T + 97iT^{2} \)
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

−11.62471807992526474996608998642, −11.20167356430647547556427443564, −9.585443783040970764194996598370, −8.685009143721553594098893964274, −8.131064361224190575895065795804, −7.36316507788016603230483295781, −6.11489922389573702566255939859, −4.93288513264412362792560894512, −3.39559758552446880762157090704, −0.61615562399452748488735250981, 2.04668242364036672859675575627, 3.02660067285236525757567046961, 4.52112332379583026210654860799, 6.27395012439851199978589508736, 7.60857074099275663763958389629, 8.578700924549002573214727864017, 9.430902215249210086180887243529, 10.25454692411286501489262866456, 11.32277503529945803399302178270, 12.02890100250707592860618402480

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