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Properties

Label 2-3840-480.29-c0-0-2
Degree $2$
Conductor $3840$
Sign $0.980 + 0.195i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 − 0.382i)3-s + (0.382 + 0.923i)5-s + (0.707 + 0.707i)9-s − i·15-s − 0.765i·17-s + (0.707 − 1.70i)19-s + (−0.541 − 0.541i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + 1.41·31-s + (−0.382 + 0.923i)45-s + 1.84i·47-s − i·49-s + (−0.292 + 0.707i)51-s + (1.30 − 0.541i)53-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (0.382 + 0.923i)5-s + (0.707 + 0.707i)9-s − i·15-s − 0.765i·17-s + (0.707 − 1.70i)19-s + (−0.541 − 0.541i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + 1.41·31-s + (−0.382 + 0.923i)45-s + 1.84i·47-s − i·49-s + (−0.292 + 0.707i)51-s + (1.30 − 0.541i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 0.980 + 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.027818015\)
\(L(\frac12)\) \(\approx\) \(1.027818015\)
\(L(1)\) 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 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 + (-0.382 - 0.923i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.707 + 0.707i)T^{2} \)
17 \( 1 + 0.765iT - T^{2} \)
19 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + (0.707 - 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 - 1.84iT - T^{2} \)
53 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - T^{2} \)
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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

−8.584787459313440418979445270737, −7.64762043759518917284821612333, −6.95146603600655717563012990180, −6.59914296966764979708311013656, −5.72000253128329412279380135224, −5.03724443199398831097224452970, −4.21133359920959626494272128446, −2.91494013625340858659717660292, −2.27597005765460890334010991359, −0.859804242788272352981378425799, 1.02268712595237959846835366755, 1.95217232128535946909203227736, 3.54651479959086670158694123612, 4.18912345665013497505418896300, 5.03861511434284852365086188556, 5.76678938330449022167569412681, 6.11654291816400124577567999078, 7.16227154753873244344752304325, 8.078598244888216836422878025552, 8.656213995692826326128036593957

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