↑

Properties

Label 2-945-105.32-c1-0-53
Degree $2$
Conductor $945$
Sign $0.999 + 0.0446i$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
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  + (2.42 + 0.650i)2-s + (3.74 + 2.16i)4-s + (−1.11 − 1.93i)5-s + (−0.654 − 2.56i)7-s + (4.12 + 4.12i)8-s + (−1.45 − 5.43i)10-s + (4.33 + 2.50i)11-s + (4.46 − 4.46i)13-s + (0.0777 − 6.65i)14-s + (3.01 + 5.22i)16-s + (−0.263 − 0.985i)17-s + (−5.51 + 3.18i)19-s + (−0.00427 − 9.66i)20-s + (8.89 + 8.89i)22-s + (−0.0712 + 0.265i)23-s + ⋯
L(s)  = 1  + (1.71 + 0.460i)2-s + (1.87 + 1.08i)4-s + (−0.500 − 0.865i)5-s + (−0.247 − 0.968i)7-s + (1.45 + 1.45i)8-s + (−0.460 − 1.71i)10-s + (1.30 + 0.754i)11-s + (1.23 − 1.23i)13-s + (0.0207 − 1.77i)14-s + (0.754 + 1.30i)16-s + (−0.0640 − 0.238i)17-s + (−1.26 + 0.729i)19-s + (−0.000955 − 2.16i)20-s + (1.89 + 1.89i)22-s + (−0.0148 + 0.0554i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0446i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.999 + 0.0446i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.16235 - 0.0929300i\)
\(L(\frac12)\) \(\approx\) \(4.16235 - 0.0929300i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.11 + 1.93i)T \)
7 \( 1 + (0.654 + 2.56i)T \)
good2 \( 1 + (-2.42 - 0.650i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-4.33 - 2.50i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.46 + 4.46i)T - 13iT^{2} \)
17 \( 1 + (0.263 + 0.985i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.51 - 3.18i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0712 - 0.265i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.05T + 29T^{2} \)
31 \( 1 + (2.12 - 3.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.349 - 1.30i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.13iT - 41T^{2} \)
43 \( 1 + (-1.22 + 1.22i)T - 43iT^{2} \)
47 \( 1 + (-7.54 - 2.02i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.0406 + 0.0108i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.17 - 8.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.02 - 1.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.6 - 3.66i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.23iT - 71T^{2} \)
73 \( 1 + (2.69 + 10.0i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (12.5 - 7.23i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.98 - 9.98i)T + 83iT^{2} \)
89 \( 1 + (2.39 + 4.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.238 - 0.238i)T + 97iT^{2} \)
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

−10.33849943300116844161489861090, −9.003470533342370029858246201621, −8.073144901470517592844424852660, −7.23332708208663274692281474701, −6.41275122980308035372200312054, −5.65578323560411942740748770620, −4.43941101752163205781510454490, −4.09470312179980032631735588519, −3.22534564680399315152516154484, −1.34968922979510088621527502199, 1.86481531879839874313329562389, 2.90939418622269309357410648570, 3.84747777592378627498151715488, 4.35719821025876809752811051235, 5.85980208580738335945936045560, 6.36011207465612723030011774563, 6.86673338872177088520520938752, 8.504003396555251722879130690995, 9.186357802143278853004488918672, 10.63121036266204675732240626195

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