↑

Properties

Label 2-2496-24.11-c1-0-79
Degree $2$
Conductor $2496$
Sign $-0.0975 + 0.995i$
Analytic cond. $19.9306$
Root an. cond. $4.46437$
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  + (1.62 + 0.609i)3-s − 1.44·5-s − 3.46i·7-s + (2.25 + 1.97i)9-s − 6.00i·11-s + i·13-s + (−2.33 − 0.878i)15-s + 4.92i·17-s + 1.73·19-s + (2.11 − 5.62i)21-s − 4.37·23-s − 2.92·25-s + (2.45 + 4.57i)27-s + 0.607·29-s − 8.83i·31-s + ⋯
L(s)  = 1  + (0.936 + 0.351i)3-s − 0.644·5-s − 1.31i·7-s + (0.752 + 0.658i)9-s − 1.81i·11-s + 0.277i·13-s + (−0.603 − 0.226i)15-s + 1.19i·17-s + 0.398·19-s + (0.461 − 1.22i)21-s − 0.912·23-s − 0.584·25-s + (0.472 + 0.881i)27-s + 0.112·29-s − 1.58i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0975 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0975 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $-0.0975 + 0.995i$
Analytic conductor: \(19.9306\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ -0.0975 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.731106529\)
\(L(\frac12)\) \(\approx\) \(1.731106529\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 0.609i)T \)
13 \( 1 - iT \)
good5 \( 1 + 1.44T + 5T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 + 6.00iT - 11T^{2} \)
17 \( 1 - 4.92iT - 17T^{2} \)
19 \( 1 - 1.73T + 19T^{2} \)
23 \( 1 + 4.37T + 23T^{2} \)
29 \( 1 - 0.607T + 29T^{2} \)
31 \( 1 + 8.83iT - 31T^{2} \)
37 \( 1 + 3.88iT - 37T^{2} \)
41 \( 1 + 7.61iT - 41T^{2} \)
43 \( 1 - 8.29T + 43T^{2} \)
47 \( 1 + 13.2T + 47T^{2} \)
53 \( 1 + 1.68T + 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 + 6.64iT - 61T^{2} \)
67 \( 1 + 5.83T + 67T^{2} \)
71 \( 1 - 5.38T + 71T^{2} \)
73 \( 1 + 4.76T + 73T^{2} \)
79 \( 1 - 0.230iT - 79T^{2} \)
83 \( 1 - 12.0iT - 83T^{2} \)
89 \( 1 - 0.471iT - 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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

−8.503556250021370073286619145259, −7.997896493828002793708259049034, −7.53767859717748972624087411826, −6.47633865759679968432602577692, −5.61947010419729629158969213582, −4.26046301759087903325979349210, −3.85542487619797849437142140852, −3.26049603804026463289515415108, −1.88319127156037292281114613985, −0.49785124601112292354177849592, 1.55482542949843620438282699077, 2.49586926171629887434648610057, 3.18713890228508161776880645783, 4.36763523497119890763102095366, 5.01191222275140652523887264041, 6.16046045554114211636826170200, 7.10110336851694600204232063682, 7.61993090637446776346613375319, 8.315427981555813077008343946250, 9.104167646954506904338696121489

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