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Properties

Label 2-3840-8.5-c1-0-63
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + i·5-s − 9-s − 4i·11-s − 6i·13-s + 15-s − 6·17-s + 4i·19-s − 25-s + i·27-s + 2i·29-s − 8·31-s − 4·33-s − 2i·37-s − 6·39-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.447i·5-s − 0.333·9-s − 1.20i·11-s − 1.66i·13-s + 0.258·15-s − 1.45·17-s + 0.917i·19-s − 0.200·25-s + 0.192i·27-s + 0.371i·29-s − 1.43·31-s − 0.696·33-s − 0.328i·37-s − 0.960·39-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: \(1\)
Selberg data: \((2,\ 3840,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 + 14iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
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

−7.912796599625954559781492977497, −7.45780800984041549694161985168, −6.42206554088495430335224426829, −5.93266186724602601673909692579, −5.25105498951903355415298803549, −4.02029321570309562791742492301, −3.17169513287499185040718865944, −2.48764007961521057354294396647, −1.22354189487460569191438815017, 0, 1.79158125388477419855318330873, 2.42346376613088842617251768509, 3.88078615245765628348069932634, 4.41252551121001951351896294295, 4.94089114047611548370176630028, 5.93975246916964310309873161845, 6.97877706390390097886219572548, 7.18617849035328307979332807699, 8.500325317326143307596410904104

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