↑

Properties

Label 2-675-135.79-c1-0-43
Degree $2$
Conductor $675$
Sign $0.969 - 0.244i$
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  + (2.36 + 0.417i)2-s + (1.71 − 0.210i)3-s + (3.54 + 1.28i)4-s + (4.15 + 0.220i)6-s + (−0.198 − 0.544i)7-s + (3.67 + 2.12i)8-s + (2.91 − 0.722i)9-s + (−2.36 − 1.98i)11-s + (6.35 + 1.47i)12-s + (−4.13 + 0.729i)13-s + (−0.241 − 1.37i)14-s + (2.03 + 1.71i)16-s + (1.72 − 0.995i)17-s + (7.18 − 0.494i)18-s + (−1.92 + 3.33i)19-s + ⋯
L(s)  = 1  + (1.67 + 0.294i)2-s + (0.992 − 0.121i)3-s + (1.77 + 0.644i)4-s + (1.69 + 0.0898i)6-s + (−0.0749 − 0.205i)7-s + (1.29 + 0.750i)8-s + (0.970 − 0.240i)9-s + (−0.714 − 0.599i)11-s + (1.83 + 0.424i)12-s + (−1.14 + 0.202i)13-s + (−0.0646 − 0.366i)14-s + (0.509 + 0.427i)16-s + (0.418 − 0.241i)17-s + (1.69 − 0.116i)18-s + (−0.441 + 0.764i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.969 - 0.244i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.969 - 0.244i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.77556 + 0.591878i\)
\(L(\frac12)\) \(\approx\) \(4.77556 + 0.591878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.210i)T \)
5 \( 1 \)
good2 \( 1 + (-2.36 - 0.417i)T + (1.87 + 0.684i)T^{2} \)
7 \( 1 + (0.198 + 0.544i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (2.36 + 1.98i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (4.13 - 0.729i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.72 + 0.995i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.92 - 3.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.52 - 4.18i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.11 - 6.30i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.55 + 0.566i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (3.49 - 2.01i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.190 + 1.07i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-4.43 + 5.28i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-1.22 - 3.37i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 - 5.40iT - 53T^{2} \)
59 \( 1 + (-7.87 + 6.61i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-12.4 + 4.51i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (8.70 - 1.53i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (0.572 + 0.991i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.169 + 0.0977i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.25 + 7.09i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (14.6 + 2.58i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (0.776 - 1.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.40 + 4.05i)T + (-16.8 - 95.5i)T^{2} \)
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

−10.61987561541468014655604073964, −9.735821450962453238317117328945, −8.594440795817301973872571819199, −7.51662381179594548823728424386, −7.07000636823314821603129149006, −5.81249152441341936754366556864, −4.98679288208558867154119370333, −3.89957430013545557220850797029, −3.14920521889217316050553749944, −2.11133180350833788202881272324, 2.25191660372632684344070064935, 2.71173058035983273849573573858, 3.99745858543960919411516850093, 4.70744890823380658494519799491, 5.61751848198151678458009726770, 6.85638197514518198095684067194, 7.61244171973043040474725457125, 8.705048280322226490856703888574, 9.879874515615261113796872269370, 10.47285399096348459153946731998

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