↑

Properties

Label 2-42e2-63.16-c1-0-31
Degree $2$
Conductor $1764$
Sign $-0.983 - 0.183i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (−1.29 − 1.15i)3-s − 1.69·5-s + (0.349 + 2.97i)9-s + 2.47·11-s + (−0.388 − 0.673i)13-s + (2.19 + 1.95i)15-s + (−1.40 − 2.43i)17-s + (2.49 − 4.31i)19-s + 0.712·23-s − 2.11·25-s + (2.97 − 4.25i)27-s + (−2.25 + 3.90i)29-s + (−2.54 + 4.41i)31-s + (−3.20 − 2.85i)33-s + (3.43 − 5.95i)37-s + ⋯
L(s)  = 1  + (−0.747 − 0.664i)3-s − 0.760·5-s + (0.116 + 0.993i)9-s + 0.746·11-s + (−0.107 − 0.186i)13-s + (0.567 + 0.505i)15-s + (−0.340 − 0.590i)17-s + (0.572 − 0.990i)19-s + 0.148·23-s − 0.422·25-s + (0.572 − 0.819i)27-s + (−0.418 + 0.725i)29-s + (−0.457 + 0.793i)31-s + (−0.558 − 0.496i)33-s + (0.565 − 0.979i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.983 - 0.183i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.983 - 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2640930401\)
\(L(\frac12)\) \(\approx\) \(0.2640930401\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.29 + 1.15i)T \)
7 \( 1 \)
good5 \( 1 + 1.69T + 5T^{2} \)
11 \( 1 - 2.47T + 11T^{2} \)
13 \( 1 + (0.388 + 0.673i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.40 + 2.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.49 + 4.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.712T + 23T^{2} \)
29 \( 1 + (2.25 - 3.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.54 - 4.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.43 + 5.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.93 - 5.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.32 + 4.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.49 + 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.944 + 1.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.14 - 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.15 + 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.99 - 6.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 + (2.49 + 4.31i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.60 - 7.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.40 - 7.63i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.82 - 8.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.32 - 7.48i)T + (-48.5 - 84.0i)T^{2} \)
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

−8.857201748029153759988953810170, −7.87570529280212131851464016140, −7.19457497629633122407348834647, −6.68820480569449432809707901353, −5.61957529952442753956864073316, −4.86588103824702198021984441636, −3.91861787381910167085290305347, −2.74063773561527040677665358032, −1.39102822775476574489128543932, −0.11913600292712082133281908758, 1.46025163921275557338660898885, 3.19907065032058515904256941259, 4.08870045045317023389044762523, 4.55349993043153232654949358494, 5.84895321399758142779550175042, 6.25675870836399516940925491470, 7.40482113621422332648740054289, 8.057744791533596182551218500520, 9.166311423962197372067493569621, 9.650280251394597681058131991349

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