↑

Properties

Label 2-968-11.5-c1-0-10
Degree $2$
Conductor $968$
Sign $-0.353 - 0.935i$
Analytic cond. $7.72951$
Root an. cond. $2.78020$
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  + (2.42 + 1.76i)3-s + (−0.927 + 2.85i)5-s + (1.61 − 1.17i)7-s + (1.85 + 5.70i)9-s + (−7.28 + 5.29i)15-s + (−1.85 + 5.70i)17-s + (−3.23 − 2.35i)19-s + 6·21-s + 23-s + (−3.23 − 2.35i)25-s + (−2.78 + 8.55i)27-s + (6.47 − 4.70i)29-s + (−2.16 − 6.65i)31-s + (1.85 + 5.70i)35-s + (0.809 − 0.587i)37-s + ⋯
L(s)  = 1  + (1.40 + 1.01i)3-s + (−0.414 + 1.27i)5-s + (0.611 − 0.444i)7-s + (0.618 + 1.90i)9-s + (−1.87 + 1.36i)15-s + (−0.449 + 1.38i)17-s + (−0.742 − 0.539i)19-s + 1.30·21-s + 0.208·23-s + (−0.647 − 0.470i)25-s + (−0.535 + 1.64i)27-s + (1.20 − 0.873i)29-s + (−0.388 − 1.19i)31-s + (0.313 + 0.964i)35-s + (0.133 − 0.0966i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.353 - 0.935i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.353 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38621 + 2.00660i\)
\(L(\frac12)\) \(\approx\) \(1.38621 + 2.00660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-2.42 - 1.76i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.927 - 2.85i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.61 + 1.17i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.85 - 5.70i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.23 + 2.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + (-6.47 + 4.70i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.16 + 6.65i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.23 + 2.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (-6.47 - 4.70i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.618 - 1.90i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.23 + 3.80i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 5T + 67T^{2} \)
71 \( 1 + (-0.927 + 2.85i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (12.9 - 9.40i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.618 - 1.90i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.618 - 1.90i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (2.16 + 6.65i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.46252725760733243788048130992, −9.435706568314044071197432819878, −8.550885681089992261697866375909, −7.930588157600893761800827905837, −7.16972612596115452434938281346, −6.07348225004211130398909240882, −4.42623737701620718282474878193, −4.05594038645738552165813442635, −2.99662220801458543063476476970, −2.16687896525309842154675076157, 1.01710887231131229252452327497, 2.07315177572463068799249620681, 3.14289928745319323238861369969, 4.40568529117200327084815735266, 5.23852238894993523173212241603, 6.65791449465686385792090293700, 7.47090513127304196756105075236, 8.263510461596096786783535450000, 8.812173980191092592260499653893, 9.156373085608611778895740132616

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