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Properties

Label 2-3840-8.5-c1-0-46
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 + 3.12·7-s − 9-s − 2i·11-s − 3.12i·13-s + 15-s + 7.12·17-s − 3.12i·19-s + 3.12i·21-s − 3.12·23-s − 25-s − i·27-s + 8.24i·29-s + 1.12·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.447i·5-s + 1.18·7-s − 0.333·9-s − 0.603i·11-s − 0.866i·13-s + 0.258·15-s + 1.72·17-s − 0.716i·19-s + 0.681i·21-s − 0.651·23-s − 0.200·25-s − 0.192i·27-s + 1.53i·29-s + 0.201·31-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

\(L(1)\) \(\approx\) \(2.137057331\)
\(L(\frac12)\) \(\approx\) \(2.137057331\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.12T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 3.12iT - 13T^{2} \)
17 \( 1 - 7.12T + 17T^{2} \)
19 \( 1 + 3.12iT - 19T^{2} \)
23 \( 1 + 3.12T + 23T^{2} \)
29 \( 1 - 8.24iT - 29T^{2} \)
31 \( 1 - 1.12T + 31T^{2} \)
37 \( 1 + 3.12iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + 4.87T + 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 - 10.2iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.400746772174218309551087024090, −7.87675521799715148211384624911, −7.08252217003396257003379187180, −5.81012061919028319135955866881, −5.33925795444934262752519669567, −4.77877457841537199152844175026, −3.73491577008777230808061747953, −3.04577624620027699710202037048, −1.75033122343477263466771348352, −0.67320592125328005663727271297, 1.25319485591704553336122760667, 1.94223810870401816827566417299, 2.94964975198539160060439988839, 4.06848572271229881634683328044, 4.75552071045085263943698514269, 5.77334949476853694547674411115, 6.30417262091288783929756915700, 7.31594718710443209074835198338, 7.86909996932907762249291567071, 8.199890765142968065415789965299

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