Properties

Label 2-3840-40.29-c1-0-74
Degree $2$
Conductor $3840$
Sign $-0.948 - 0.316i$
Analytic cond. $30.6625$
Root an. cond. $5.53737$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

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

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

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $-0.948 - 0.316i$
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: \(1\)
Selberg data: \((2,\ 3840,\ (\ :1/2),\ -0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−7.67100691982592494319962475267, −7.18213666900942543000580526143, −7.07665093602553087222345303659, −5.87723075347834717108006791312, −4.88361477114835320659974691443, −4.18130388667794124337835390943, −3.70967122958870468544864992828, −2.46040909099634949532930674713, −1.09743047650742575618685524963, 0, 1.32837498739201785889138248628, 2.60520460625860638316609689475, 3.53563637696182971539412512751, 4.36734315782402222249206983174, 5.39548168780312930976396048678, 5.72215104117866980872136179120, 6.55695265434507127195503621243, 7.47767137471417224354679673393, 8.407437464506144848470774426634

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