Properties

Label 2-3840-8.5-c1-0-34
Degree $2$
Conductor $3840$
Sign $0.707 - 0.707i$
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 + i·5-s + 4·7-s − 9-s + 2i·13-s − 15-s + 6·17-s − 4i·19-s + 4i·21-s − 25-s i·27-s − 6i·29-s + 8·31-s + 4i·35-s − 2i·37-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.447i·5-s + 1.51·7-s − 0.333·9-s + 0.554i·13-s − 0.258·15-s + 1.45·17-s − 0.917i·19-s + 0.872i·21-s − 0.200·25-s − 0.192i·27-s − 1.11i·29-s + 1.43·31-s + 0.676i·35-s − 0.328i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(30.6625\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (1921, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.513961194\)
\(L(\frac12)\) \(\approx\) \(2.513961194\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.421374187747940337034609501809, −7.953902831451915862831395363159, −7.26069398724145297566384399809, −6.26946337792334902790253549379, −5.49899491284154989945564468472, −4.69294558814507026063417205080, −4.18193085068625208485071899012, −3.07209528420726367066002337904, −2.19627369630982448406835930496, −1.01473690259921246217950180308, 0.995658404420836166060860145934, 1.58773439016140608985161131265, 2.73657141623597950703662076789, 3.78401217892843558689289440772, 4.78170539601653697535486997058, 5.40475459459180967659541159039, 6.01307665431196708029310950435, 7.14699917627277091601017031436, 7.83108482472909831459656684776, 8.215407454374943375546659803878

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