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 + 6·13-s + (−1 + 2i)15-s + 2i·17-s + 2i·21-s + 4i·23-s + (−3 − 4i)25-s + 27-s + 8·31-s + 2i·33-s + (−4 − 2i)35-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 + 1.66·13-s + (−0.258 + 0.516i)15-s + 0.485i·17-s + 0.436i·21-s + 0.834i·23-s + (−0.600 − 0.800i)25-s + 0.192·27-s + 1.43·31-s + 0.348i·33-s + (−0.676 − 0.338i)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: $-0.316 - 0.948i$
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

\(L(1)\) \(\approx\) \(2.211574060\)
\(L(\frac12)\) \(\approx\) \(2.211574060\)
\(L(\frac{3}{2})\) 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 \( 1 - 6T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
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

−8.513275540089964195522773002856, −8.108102182958424751085592907644, −7.33152834929353575735137745207, −6.40954027438300541165022039644, −6.02668805031417311521054762534, −4.83117280368541482849724808824, −3.88503491778614909349831885469, −3.27607771599875131833892687277, −2.41968875467660309684656623631, −1.40458780056428103856653163408, 0.64980809887545010900785055522, 1.42825744328556809689928239227, 2.84530438744157926399235612444, 3.74271592482130219761233705900, 4.25471601595203104119118644008, 5.13410895862985464579805844317, 6.11728987448086918048478547519, 6.83856423514350708286463306376, 7.75466662178013593451115977926, 8.443956919788316230091552925038

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