↑

Properties

Label 2-960-120.29-c2-0-0
Degree $2$
Conductor $960$
Sign $-0.972 + 0.232i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.90 − 0.751i)3-s + (−1.38 + 4.80i)5-s + 3.03i·7-s + (7.86 − 4.36i)9-s − 11.4·11-s − 16.9·13-s + (−0.413 + 14.9i)15-s − 18.1·17-s − 16.9i·19-s + (2.28 + 8.82i)21-s − 22.3·23-s + (−21.1 − 13.3i)25-s + (19.5 − 18.5i)27-s − 17.8·29-s + 16.9·31-s + ⋯
L(s)  = 1  + (0.968 − 0.250i)3-s + (−0.277 + 0.960i)5-s + 0.434i·7-s + (0.874 − 0.485i)9-s − 1.04·11-s − 1.30·13-s + (−0.0275 + 0.999i)15-s − 1.06·17-s − 0.891i·19-s + (0.108 + 0.420i)21-s − 0.971·23-s + (−0.846 − 0.532i)25-s + (0.724 − 0.688i)27-s − 0.617·29-s + 0.547·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.972 + 0.232i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ -0.972 + 0.232i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.05494112644\)
\(L(\frac12)\) \(\approx\) \(0.05494112644\)
\(L(2)\) 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 \( 1 + (-2.90 + 0.751i)T \)
5 \( 1 + (1.38 - 4.80i)T \)
good7 \( 1 - 3.03iT - 49T^{2} \)
11 \( 1 + 11.4T + 121T^{2} \)
13 \( 1 + 16.9T + 169T^{2} \)
17 \( 1 + 18.1T + 289T^{2} \)
19 \( 1 + 16.9iT - 361T^{2} \)
23 \( 1 + 22.3T + 529T^{2} \)
29 \( 1 + 17.8T + 841T^{2} \)
31 \( 1 - 16.9T + 961T^{2} \)
37 \( 1 + 1.21T + 1.36e3T^{2} \)
41 \( 1 + 42.0iT - 1.68e3T^{2} \)
43 \( 1 - 1.98T + 1.84e3T^{2} \)
47 \( 1 - 52.4T + 2.20e3T^{2} \)
53 \( 1 - 8.58iT - 2.80e3T^{2} \)
59 \( 1 + 93.0T + 3.48e3T^{2} \)
61 \( 1 + 80.4iT - 3.72e3T^{2} \)
67 \( 1 + 38.1T + 4.48e3T^{2} \)
71 \( 1 - 109. iT - 5.04e3T^{2} \)
73 \( 1 + 28.2iT - 5.32e3T^{2} \)
79 \( 1 + 56.1T + 6.24e3T^{2} \)
83 \( 1 - 126. iT - 6.88e3T^{2} \)
89 \( 1 + 126. iT - 7.92e3T^{2} \)
97 \( 1 - 153. iT - 9.40e3T^{2} \)
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

−10.20896556998370574167531654222, −9.417388107274485139754874109280, −8.570949212601294633657535192171, −7.61278766932951756059698914500, −7.20353990876244593167235009777, −6.19997236931690160669800506389, −4.91759298885708217553671738699, −3.86956471515919349540705309459, −2.55733207392541373262833440419, −2.35944562327085523941812039921, 0.01363764986540713015037309562, 1.77386006170818698914556228362, 2.82407568229091782213436771874, 4.15724182517063448294100157860, 4.65472457322289540210446741830, 5.74218793743644444318862278783, 7.24564084616584418651280209050, 7.80878967946308741705920223800, 8.486106443359763542542101265890, 9.384893884618156873724682428075

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