Properties

Label 2-3840-40.29-c1-0-27
Degree $2$
Conductor $3840$
Sign $0.316 - 0.948i$
Analytic cond. $30.6625$
Root an. cond. $5.53737$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (2 − i)5-s + 4i·7-s + 9-s + 4i·11-s + (−2 + i)15-s − 4i·17-s − 4i·21-s − 4i·23-s + (3 − 4i)25-s − 27-s + 6i·29-s + 4·31-s − 4i·33-s + (4 + 8i)35-s + ⋯
L(s)  = 1  − 0.577·3-s + (0.894 − 0.447i)5-s + 1.51i·7-s + 0.333·9-s + 1.20i·11-s + (−0.516 + 0.258i)15-s − 0.970i·17-s − 0.872i·21-s − 0.834i·23-s + (0.600 − 0.800i)25-s − 0.192·27-s + 1.11i·29-s + 0.718·31-s − 0.696i·33-s + (0.676 + 1.35i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.316 - 0.948i$
Analytic conductor: \(30.6625\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.774280956\)
\(L(\frac12)\) \(\approx\) \(1.774280956\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + (-2 + i)T \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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

−8.849374337804467343214093047852, −7.929210961192851874185850441225, −7.01765088310070113605281841117, −6.21814891903146690481519422026, −5.70920482603491429739545368637, −4.88304276254778416050683265833, −4.48843092112184210901224627671, −2.76640496182648206946222259235, −2.27108252912912709956622885863, −1.13379911430019667683309605047, 0.63134009277517368460804155646, 1.54984243871760249027867512690, 2.84455365631458794571511144179, 3.78725566099458640682077395931, 4.46442796500013484512843665417, 5.58998272192695646335070045087, 6.15116493275716350907595604770, 6.65894874970356530470862956954, 7.63731906834285617515929822828, 8.100339686624105484240114584092

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