Properties

Label 2-3840-40.29-c1-0-48
Degree $2$
Conductor $3840$
Sign $0.948 - 0.316i$
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  + 3-s + (1 − 2i)5-s + 2i·7-s + 9-s + 6i·11-s + 2·13-s + (1 − 2i)15-s − 6i·17-s − 4i·19-s + 2i·21-s + 8i·23-s + (−3 − 4i)25-s + 27-s + 8·31-s + 6i·33-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.447 − 0.894i)5-s + 0.755i·7-s + 0.333·9-s + 1.80i·11-s + 0.554·13-s + (0.258 − 0.516i)15-s − 1.45i·17-s − 0.917i·19-s + 0.436i·21-s + 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.192·27-s + 1.43·31-s + 1.04i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.948 - 0.316i$
Analytic conductor: \(30.6625\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :1/2),\ 0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.723753614\)
\(L(\frac12)\) \(\approx\) \(2.723753614\)
\(L(\frac{3}{2})\) 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 \( 1 - 6iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
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

−8.733820273216914718701308146871, −7.77353422828751937920084895569, −7.21551074569229676937747824172, −6.33824499941308818949989329903, −5.28696956116940240275729054352, −4.86428778680526759388612534048, −4.03516442239569406157107829835, −2.76891566852514825993641777777, −2.12576054785698034647156900768, −1.10440756254888688392528701383, 0.854119067424763064850588944514, 2.02482757013089088582550482867, 3.09016986958976323889115787108, 3.62159602338295760947142595201, 4.39189430287182709092506074615, 5.84875309917937703041865679390, 6.17459060460245651449405104106, 6.88999540728944346452539273545, 7.920632073914627816261596715370, 8.382924299285156204049586835220

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