Properties

Degree $2$
Conductor $3840$
Sign $-0.316 - 0.948i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (1 − 2i)5-s − 2i·7-s + 9-s + 2i·11-s + 2·13-s + (−1 + 2i)15-s + 6i·17-s + 8i·19-s + 2i·21-s + 4i·23-s + (−3 − 4i)25-s − 27-s − 8i·29-s − 2i·33-s + ⋯
L(s)  = 1  − 0.577·3-s + (0.447 − 0.894i)5-s − 0.755i·7-s + 0.333·9-s + 0.603i·11-s + 0.554·13-s + (−0.258 + 0.516i)15-s + 1.45i·17-s + 1.83i·19-s + 0.436i·21-s + 0.834i·23-s + (−0.600 − 0.800i)25-s − 0.192·27-s − 1.48i·29-s − 0.348i·33-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: $-0.316 - 0.948i$
Motivic weight: \(1\)
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

\(L(1)\) \(\approx\) \(0.6820523762\)
\(L(\frac12)\) \(\approx\) \(0.6820523762\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
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

−8.561753271073022128325897227349, −8.054583092796113678871690138821, −7.26665600984162297707491558447, −6.23523162061230787235868153417, −5.85395243289108653291132987370, −4.98719865974542257406460526183, −4.11091320673838646542904295556, −3.61461112464794846180438201666, −1.78773574803789198224948813380, −1.37645479050869831430177493519, 0.20946061459046152535788349015, 1.70187213293398571956481567266, 2.84451697234790793650844228161, 3.28607537413914765863373658962, 4.78381778627784541015896606327, 5.25039049660556180526269115043, 6.09899961831971042395416080145, 6.85242489540295407919268452300, 7.13189537479291351882587072422, 8.449617658935682980101228900197

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