↑

Properties

Label 2-1224-1224.859-c0-0-1
Degree $2$
Conductor $1224$
Sign $0.999 + 0.0407i$
Analytic cond. $0.610855$
Root an. cond. $0.781572$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 − 0.5i)3-s + (0.866 + 0.499i)4-s + (0.965 − 0.258i)6-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−1.25 − 0.965i)11-s + 12-s + (0.500 + 0.866i)16-s + (−0.965 − 0.258i)17-s + (0.707 − 0.707i)18-s + (−1.22 + 1.22i)19-s + (−0.965 − 1.25i)22-s + (0.965 + 0.258i)24-s + (−0.258 + 0.965i)25-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 − 0.5i)3-s + (0.866 + 0.499i)4-s + (0.965 − 0.258i)6-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−1.25 − 0.965i)11-s + 12-s + (0.500 + 0.866i)16-s + (−0.965 − 0.258i)17-s + (0.707 − 0.707i)18-s + (−1.22 + 1.22i)19-s + (−0.965 − 1.25i)22-s + (0.965 + 0.258i)24-s + (−0.258 + 0.965i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.999 + 0.0407i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.999 + 0.0407i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.300050876\)
\(L(\frac12)\) \(\approx\) \(2.300050876\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.965 + 0.258i)T \)
good5 \( 1 + (0.258 - 0.965i)T^{2} \)
7 \( 1 + (0.258 + 0.965i)T^{2} \)
11 \( 1 + (1.25 + 0.965i)T + (0.258 + 0.965i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
23 \( 1 + (0.965 + 0.258i)T^{2} \)
29 \( 1 + (-0.965 + 0.258i)T^{2} \)
31 \( 1 + (-0.258 + 0.965i)T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (-1.57 - 0.207i)T + (0.965 + 0.258i)T^{2} \)
43 \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.258 + 0.965i)T^{2} \)
67 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.0999 + 0.241i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.258 - 0.965i)T^{2} \)
83 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (-1.20 + 0.158i)T + (0.965 - 0.258i)T^{2} \)
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

−9.952898369497892917823297693299, −8.763108641306558879233796902406, −8.094268200028574663552163648208, −7.53247826848947190699532754487, −6.47835162007134266125345026751, −5.85306108852327144768184695134, −4.70770670555130353528403102948, −3.69955911028254979234470266749, −2.84715385184354592229401556803, −1.91713421047311612487806105333, 2.25934851205809238318907065960, 2.54443935499194259395267591006, 4.02367012838623944818791291820, 4.54843712757463332491035130994, 5.38819159267613979891882461615, 6.63777589959428885291892667808, 7.37234348197125593690793608318, 8.295310443584580984709585789386, 9.190924113225765314553474734679, 10.19917127988427247033742509359

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