↑

Properties

Label 2-675-25.16-c1-0-38
Degree $2$
Conductor $675$
Sign $-0.0871 + 0.996i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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  + (1.18 + 0.864i)2-s + (0.0501 + 0.154i)4-s + (0.976 − 2.01i)5-s − 4.82·7-s + (0.835 − 2.57i)8-s + (2.90 − 1.54i)10-s + (−4.15 − 3.02i)11-s + (−1.44 + 1.04i)13-s + (−5.74 − 4.17i)14-s + (3.47 − 2.52i)16-s + (−1.74 + 5.37i)17-s + (2.40 − 7.39i)19-s + (0.359 + 0.0498i)20-s + (−2.33 − 7.19i)22-s + (1.55 + 1.12i)23-s + ⋯
L(s)  = 1  + (0.841 + 0.611i)2-s + (0.0250 + 0.0770i)4-s + (0.436 − 0.899i)5-s − 1.82·7-s + (0.295 − 0.908i)8-s + (0.917 − 0.489i)10-s + (−1.25 − 0.911i)11-s + (−0.400 + 0.291i)13-s + (−1.53 − 1.11i)14-s + (0.869 − 0.631i)16-s + (−0.423 + 1.30i)17-s + (0.551 − 1.69i)19-s + (0.0802 + 0.0111i)20-s + (−0.498 − 1.53i)22-s + (0.324 + 0.235i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.0871 + 0.996i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.0871 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.921482 - 1.00564i\)
\(L(\frac12)\) \(\approx\) \(0.921482 - 1.00564i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.976 + 2.01i)T \)
good2 \( 1 + (-1.18 - 0.864i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + 4.82T + 7T^{2} \)
11 \( 1 + (4.15 + 3.02i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.44 - 1.04i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.74 - 5.37i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.40 + 7.39i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.55 - 1.12i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.879 - 2.70i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.452 - 1.39i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.75 + 3.45i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.65 + 1.93i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.72T + 43T^{2} \)
47 \( 1 + (1.35 + 4.18i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.241 + 0.742i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.46 + 3.97i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.382 - 0.278i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.64 + 8.13i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.20 - 6.77i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.92 - 3.58i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.48 + 10.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.69 - 5.21i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (9.36 + 6.80i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.969 - 2.98i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.11348633907960080641998380263, −9.413000968891044663976861824442, −8.662040720836592099640401993098, −7.30963532449627257930040400850, −6.43716717626398131359538966743, −5.73253462044671934582119629706, −5.00315717434424316532090062052, −3.85330815084568221750684948227, −2.71612157638814738584589676515, −0.51744853206733789114292583226, 2.54355168499650655006211207933, 2.86346381096151608571447208960, 3.97893004700434385842150580228, 5.24474057670228908898535940685, 6.09048703009282318648169250611, 7.15757946787478755201872844737, 7.86056796718474304407538300856, 9.501993379082380636187283646332, 9.956895179132718342687761269800, 10.67230336837054892559713557764

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