↑

Properties

Label 2-384-12.11-c5-0-47
Degree $2$
Conductor $384$
Sign $0.760 - 0.649i$
Analytic cond. $61.5873$
Root an. cond. $7.84776$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (10.1 + 11.8i)3-s − 19.3i·5-s + 82.7i·7-s + (−38.1 + 239. i)9-s + 332.·11-s + 1.17e3·13-s + (229. − 195. i)15-s − 1.98e3i·17-s − 1.34e3i·19-s + (−981. + 837. i)21-s − 1.83e3·23-s + 2.75e3·25-s + (−3.23e3 + 1.97e3i)27-s − 1.92e3i·29-s + 5.80e3i·31-s + ⋯
L(s)  = 1  + (0.649 + 0.760i)3-s − 0.345i·5-s + 0.638i·7-s + (−0.157 + 0.987i)9-s + 0.828·11-s + 1.92·13-s + (0.262 − 0.224i)15-s − 1.66i·17-s − 0.855i·19-s + (−0.485 + 0.414i)21-s − 0.722·23-s + 0.880·25-s + (−0.853 + 0.521i)27-s − 0.424i·29-s + 1.08i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.760 - 0.649i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ 0.760 - 0.649i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.211876372\)
\(L(\frac12)\) \(\approx\) \(3.211876372\)
\(L(\frac{7}{2})\) 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 \( 1 + (-10.1 - 11.8i)T \)
good5 \( 1 + 19.3iT - 3.12e3T^{2} \)
7 \( 1 - 82.7iT - 1.68e4T^{2} \)
11 \( 1 - 332.T + 1.61e5T^{2} \)
13 \( 1 - 1.17e3T + 3.71e5T^{2} \)
17 \( 1 + 1.98e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.34e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.83e3T + 6.43e6T^{2} \)
29 \( 1 + 1.92e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.80e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.80e3T + 6.93e7T^{2} \)
41 \( 1 + 4.15e3iT - 1.15e8T^{2} \)
43 \( 1 + 8.37e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.80e3T + 2.29e8T^{2} \)
53 \( 1 - 2.27e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.86e4T + 7.14e8T^{2} \)
61 \( 1 + 1.95e4T + 8.44e8T^{2} \)
67 \( 1 - 5.09e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.98e4T + 1.80e9T^{2} \)
73 \( 1 - 3.70e4T + 2.07e9T^{2} \)
79 \( 1 - 5.05e4iT - 3.07e9T^{2} \)
83 \( 1 - 9.63e4T + 3.93e9T^{2} \)
89 \( 1 + 1.79e4iT - 5.58e9T^{2} \)
97 \( 1 - 4.83e4T + 8.58e9T^{2} \)
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

−10.65393084825199600868380769069, −9.311138715222758333692350953921, −9.017602498389535735924390220317, −8.177490392394690010551263354483, −6.84038025661554592524387781385, −5.65860672910766334379619848529, −4.62768849707595694277308588538, −3.60067570036998634145520073623, −2.52117687399235089061601787913, −1.01450713240615435649824850453, 0.985296476621003394371106370965, 1.81290652705135816358264820094, 3.49308916705555056640003250925, 3.98203131948412192925033583210, 6.19116405410700219063319550679, 6.39833079193944569231009056546, 7.79420694846689361132264485864, 8.376246615700375709732095187588, 9.347354397897248879641449659757, 10.51882339489369844388000429235

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