↑

Properties

Label 2-975-65.33-c1-0-16
Degree $2$
Conductor $975$
Sign $0.145 - 0.989i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + (−0.258 + 0.448i)2-s + (0.965 − 0.258i)3-s + (0.866 + 1.5i)4-s + (−0.133 + 0.5i)6-s − 1.93·8-s + (0.866 − 0.499i)9-s + (−0.964 − 3.59i)11-s + (1.22 + 1.22i)12-s + (0.0693 + 3.60i)13-s + (−1.23 + 2.13i)16-s + (−1.08 + 4.05i)17-s + 0.517i·18-s + (7.09 + 1.90i)19-s + (1.86 + 0.499i)22-s + (0.965 + 3.60i)23-s + (−1.86 + 0.499i)24-s + ⋯
L(s)  = 1  + (−0.183 + 0.316i)2-s + (0.557 − 0.149i)3-s + (0.433 + 0.750i)4-s + (−0.0546 + 0.204i)6-s − 0.683·8-s + (0.288 − 0.166i)9-s + (−0.290 − 1.08i)11-s + (0.353 + 0.353i)12-s + (0.0192 + 0.999i)13-s + (−0.308 + 0.533i)16-s + (−0.263 + 0.983i)17-s + 0.122i·18-s + (1.62 + 0.436i)19-s + (0.397 + 0.106i)22-s + (0.201 + 0.751i)23-s + (−0.380 + 0.102i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.145 - 0.989i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.145 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40488 + 1.21332i\)
\(L(\frac12)\) \(\approx\) \(1.40488 + 1.21332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
13 \( 1 + (-0.0693 - 3.60i)T \)
good2 \( 1 + (0.258 - 0.448i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.964 + 3.59i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.08 - 4.05i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-7.09 - 1.90i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.965 - 3.60i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-7.56 - 4.36i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 - 3i)T + 31iT^{2} \)
37 \( 1 + (-4.69 - 2.70i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.46 - 1.19i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.84 + 2.63i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 1.41iT - 47T^{2} \)
53 \( 1 + (5.27 + 5.27i)T + 53iT^{2} \)
59 \( 1 + (-2.36 + 8.83i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.598 - 1.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.22 + 2.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.66 + 9.96i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 0.896T + 73T^{2} \)
79 \( 1 + 8.73iT - 79T^{2} \)
83 \( 1 - 9.76iT - 83T^{2} \)
89 \( 1 + (-10.8 + 2.90i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.637 - 1.10i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.04283434224763787300302978232, −9.122432290002511601464485091177, −8.351314092513679043360504247238, −7.890959987858003081301993422859, −6.83662482645292238989686827985, −6.25226719831738402856976625165, −4.96428464840322421840660532540, −3.55846809911113864690807153629, −3.05541802197015034301182414617, −1.59247100229562465509216860851, 0.898171243636198607889065202859, 2.40326933378123300095527589828, 3.03446918374687929428661672079, 4.64580580565461789411551227695, 5.31139262546271870603665289052, 6.52490806984767103701095941076, 7.32747715520557256810135980183, 8.171883245931789761154383323133, 9.269665313789425971467931507231, 9.934086678327872473433355889753

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