↑

Properties

Label 2-264-11.5-c1-0-4
Degree $2$
Conductor $264$
Sign $0.917 + 0.398i$
Analytic cond. $2.10805$
Root an. cond. $1.45191$
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  + (0.809 + 0.587i)3-s + (0.773 − 2.38i)5-s + (2.27 − 1.65i)7-s + (0.309 + 0.951i)9-s + (−2.77 + 1.81i)11-s + (−1.22 − 3.78i)13-s + (2.02 − 1.47i)15-s + (0.175 − 0.539i)17-s + (5.11 + 3.71i)19-s + 2.81·21-s + 2.88·23-s + (−1.02 − 0.744i)25-s + (−0.309 + 0.951i)27-s + (−5.90 + 4.28i)29-s + (2.84 + 8.76i)31-s + ⋯
L(s)  = 1  + (0.467 + 0.339i)3-s + (0.345 − 1.06i)5-s + (0.860 − 0.625i)7-s + (0.103 + 0.317i)9-s + (−0.837 + 0.546i)11-s + (−0.341 − 1.04i)13-s + (0.522 − 0.379i)15-s + (0.0425 − 0.130i)17-s + (1.17 + 0.851i)19-s + 0.614·21-s + 0.601·23-s + (−0.205 − 0.148i)25-s + (−0.0594 + 0.183i)27-s + (−1.09 + 0.796i)29-s + (0.511 + 1.57i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.917 + 0.398i$
Analytic conductor: \(2.10805\)
Root analytic conductor: \(1.45191\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :1/2),\ 0.917 + 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55199 - 0.322543i\)
\(L(\frac12)\) \(\approx\) \(1.55199 - 0.322543i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (2.77 - 1.81i)T \)
good5 \( 1 + (-0.773 + 2.38i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.27 + 1.65i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.22 + 3.78i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.175 + 0.539i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.11 - 3.71i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.88T + 23T^{2} \)
29 \( 1 + (5.90 - 4.28i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.84 - 8.76i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.283 + 0.206i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (8.45 + 6.14i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.11T + 43T^{2} \)
47 \( 1 + (3.89 + 2.82i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.46 - 4.50i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.86 - 4.26i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.02 + 12.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + (1.31 - 4.04i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.335 - 0.243i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.59 - 4.91i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.09 + 3.36i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + (-3.72 - 11.4i)T + (-78.4 + 57.0i)T^{2} \)
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

−12.05663980737399751584088337404, −10.68882652634047754694621030258, −10.04925744588593600565072645801, −8.980417031063491473313396298365, −8.046430134599716354685979666361, −7.31519712964926434037816094371, −5.23272561450408664480816576438, −4.96377195707303908578908125614, −3.31699102767595048564678666933, −1.49745843061272042942271829634, 2.09650556352143332783752262514, 3.09677372506085918750248703005, 4.86782922366922594300299520152, 6.09452138175938389886059588982, 7.18838343153766252095665688812, 8.043629495321263226188915106139, 9.138029844092159390930592952474, 10.07603653021967234487671929293, 11.35257864215343848526338256650, 11.66656967504868137673671162888

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