Properties

Label 2-3840-40.29-c1-0-77
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (2 − i)5-s − 4i·7-s + 9-s + 4·13-s + (−2 + i)15-s − 8i·19-s + 4i·21-s − 4i·23-s + (3 − 4i)25-s − 27-s + 6i·29-s + 8·31-s + (−4 − 8i)35-s − 4·37-s + ⋯
L(s)  = 1  − 0.577·3-s + (0.894 − 0.447i)5-s − 1.51i·7-s + 0.333·9-s + 1.10·13-s + (−0.516 + 0.258i)15-s − 1.83i·19-s + 0.872i·21-s − 0.834i·23-s + (0.600 − 0.800i)25-s − 0.192·27-s + 1.11i·29-s + 1.43·31-s + (−0.676 − 1.35i)35-s − 0.657·37-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

\(L(1)\) \(\approx\) \(1.907093476\)
\(L(\frac12)\) \(\approx\) \(1.907093476\)
\(L(\frac{3}{2})\) 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 \( 1 - 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 6iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.439280947041724828369778986336, −7.28467451916425409737504924828, −6.71883504358325432517094108402, −6.19449141239918099876392670374, −5.12563768601513364714137869726, −4.62351507821192938027965605164, −3.77837432412368337841722462532, −2.63868692538247275934287778627, −1.30946335159695553694491586273, −0.66490024439535186336475027711, 1.38670433189134003176200789978, 2.18095538367954224802715980440, 3.15912533813683896638130809239, 4.11529057434854098221539106148, 5.39294326186920822235473980826, 5.76603369735828493651123605100, 6.19659691040763031589308609790, 7.03867932419971488116179221426, 8.174774246779822126799057410626, 8.643341868131511253919080652131

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