Properties

Degree $2$
Conductor $3840$
Sign $-0.707 - 0.707i$
Motivic weight $1$
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: $-0.707 - 0.707i$
Motivic weight: \(1\)
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

\(L(1)\) \(\approx\) \(0.5791960556\)
\(L(\frac12)\) \(\approx\) \(0.5791960556\)
\(L(\frac{3}{2})\) 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

−9.049501633708895475355200817898, −8.074728951994574644427749423187, −7.27986764413298653869240839105, −6.51345380999056069954935343856, −5.62046007723335698577820786912, −5.18374109706472440891985596441, −4.03695833279852709258724400268, −3.33838303718534531046498251516, −2.69592201268720051595776666302, −1.06792132574825362115703732758, 0.18873018619139776894527362869, 1.60093197994726877772893579022, 2.70367902591892400586404656081, 3.44081796995605314524539879977, 4.13367776901290376629116522757, 5.69204567733969071922162444918, 5.89931564055475593586577752845, 6.81715580327141751355009472148, 7.36781763279993281526828451162, 8.041281451502253573779367608135

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