↑

Properties

Label 2-24-8.5-c1-0-1
Degree $2$
Conductor $24$
Sign $0.707 + 0.707i$
Analytic cond. $0.191640$
Root an. cond. $0.437768$
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 − i)2-s − i·3-s + 2i·4-s + 2i·5-s + (−1 + i)6-s − 2·7-s + (2 − 2i)8-s − 9-s + (2 − 2i)10-s + 2·12-s − 4i·13-s + (2 + 2i)14-s + 2·15-s − 4·16-s − 2·17-s + (1 + i)18-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s − 0.577i·3-s + i·4-s + 0.894i·5-s + (−0.408 + 0.408i)6-s − 0.755·7-s + (0.707 − 0.707i)8-s − 0.333·9-s + (0.632 − 0.632i)10-s + 0.577·12-s − 1.10i·13-s + (0.534 + 0.534i)14-s + 0.516·15-s − 16-s − 0.485·17-s + (0.235 + 0.235i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(0.191640\)
Root analytic conductor: \(0.437768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.437223 - 0.181103i\)
\(L(\frac12)\) \(\approx\) \(0.437223 - 0.181103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + iT \)
good5 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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

−18.00693396794925179402267911549, −16.83562473117415672804824087885, −15.27538700913625290547921376262, −13.50708743348838688925759206637, −12.43052393866941433087292838450, −11.00575814520349641732290778350, −9.848579168978871689408502955344, −8.076922677812066366448595950498, −6.63334300779775235394940480185, −3.03882077536188858522288571892, 4.87562501130936063565193053851, 6.73600798783002870211285003472, 8.763614598696003267997171626681, 9.529905838361445048625055372990, 11.14776256340104007773522735073, 13.06770714935972084475521959982, 14.57354599873087224304882324985, 15.98223877781209033394422032748, 16.49123228869151546957401212453, 17.66669516292985138000552330727

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