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Properties

Label 2-3840-5.4-c1-0-30
Degree $2$
Conductor $3840$
Sign $0.832 - 0.554i$
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  − i·3-s + (1.23 + 1.86i)5-s − 0.746i·7-s − 9-s − 5.36·11-s − 2.92i·13-s + (1.86 − 1.23i)15-s − 2.13i·17-s − 1.73·19-s − 0.746·21-s + 7.49i·23-s + (−1.92 + 4.61i)25-s + i·27-s + 6.74·29-s − 2.64·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.554 + 0.832i)5-s − 0.282i·7-s − 0.333·9-s − 1.61·11-s − 0.811i·13-s + (0.480 − 0.320i)15-s − 0.517i·17-s − 0.397·19-s − 0.162·21-s + 1.56i·23-s + (−0.385 + 0.922i)25-s + 0.192i·27-s + 1.25·29-s − 0.475·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.832 - 0.554i$
Analytic conductor: \(30.6625\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :1/2),\ 0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.587588545\)
\(L(\frac12)\) \(\approx\) \(1.587588545\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-1.23 - 1.86i)T \)
good7 \( 1 + 0.746iT - 7T^{2} \)
11 \( 1 + 5.36T + 11T^{2} \)
13 \( 1 + 2.92iT - 13T^{2} \)
17 \( 1 + 2.13iT - 17T^{2} \)
19 \( 1 + 1.73T + 19T^{2} \)
23 \( 1 - 7.49iT - 23T^{2} \)
29 \( 1 - 6.74T + 29T^{2} \)
31 \( 1 + 2.64T + 31T^{2} \)
37 \( 1 - 1.07iT - 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 - 7.44iT - 43T^{2} \)
47 \( 1 - 1.73iT - 47T^{2} \)
53 \( 1 + 7.72iT - 53T^{2} \)
59 \( 1 - 6.85T + 59T^{2} \)
61 \( 1 - 6.45T + 61T^{2} \)
67 \( 1 - 7.44iT - 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 0.690iT - 73T^{2} \)
79 \( 1 - 2.64T + 79T^{2} \)
83 \( 1 - 5.85iT - 83T^{2} \)
89 \( 1 - 7.59T + 89T^{2} \)
97 \( 1 - 14.1iT - 97T^{2} \)
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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

−8.268710257123871771289084377522, −7.74494510705900815263584247446, −7.18302123577696449667707085331, −6.39211872191759277852549168741, −5.55549199077952524607445740863, −5.11412412114249031630199803951, −3.73807944712469764707858735419, −2.78233545243387307025533445105, −2.33350357684471753508456861363, −0.936327840889456866961341720190, 0.54503343680930240106742410396, 2.12086212810733459352710565878, 2.65813452060761087784508722412, 4.05424230186111439383508386541, 4.64540646801825136009786517978, 5.37800319391986445507452143551, 5.99171799663352178438285248704, 6.85710774315018971233849948509, 7.966189007887019104136848260532, 8.520560584502095781770105454357

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