↑

Properties

Label 2-245-35.4-c1-0-7
Degree $2$
Conductor $245$
Sign $-0.441 + 0.897i$
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  + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (−0.500 + 0.866i)4-s + (−1.23 + 1.86i)5-s + 0.999·6-s − 3i·8-s + (−1 − 1.73i)9-s + (0.133 − 2.23i)10-s + (0.866 − 0.499i)12-s − 2i·13-s + (2 − i)15-s + (0.500 + 0.866i)16-s + (−1.73 − i)17-s + (1.73 + i)18-s + (−3 − 5.19i)19-s + (−0.999 − 2i)20-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (−0.250 + 0.433i)4-s + (−0.550 + 0.834i)5-s + 0.408·6-s − 1.06i·8-s + (−0.333 − 0.577i)9-s + (0.0423 − 0.705i)10-s + (0.249 − 0.144i)12-s − 0.554i·13-s + (0.516 − 0.258i)15-s + (0.125 + 0.216i)16-s + (−0.420 − 0.242i)17-s + (0.408 + 0.235i)18-s + (−0.688 − 1.19i)19-s + (−0.223 − 0.447i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.441 + 0.897i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.441 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0754839 - 0.121205i\)
\(L(\frac12)\) \(\approx\) \(0.0754839 - 0.121205i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.23 - 1.86i)T \)
7 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.92 + 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - 7iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.33 + 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (5.19 + 3i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11iT - 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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

−11.64231950683857230360666832978, −10.96929848247806857700948469271, −9.726863145790024520807496131507, −8.771930599743764200590183228898, −7.71344682988807323143604297603, −6.94839155431265055827884772331, −6.01838284493234335911771964454, −4.25236468525694142881114824478, −3.00963034428943585660612534693, −0.14531022732824240318686317720, 1.83895907454852448713953458186, 4.15103403855096929517846152514, 5.11288436494430705583372029567, 6.11845769572268643268179993261, 7.906753655950529321518260937554, 8.592241243058548382250794644736, 9.577849733230407296176285489547, 10.52522760788560324126306156583, 11.30104632370477761847189495058, 12.08286456654639250960919158980

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