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Properties

Label 2-3840-8.5-c1-0-28
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 − 5.12·7-s − 9-s − 2i·11-s + 5.12i·13-s + 15-s − 1.12·17-s + 5.12i·19-s − 5.12i·21-s + 5.12·23-s − 25-s − i·27-s − 8.24i·29-s − 7.12·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.447i·5-s − 1.93·7-s − 0.333·9-s − 0.603i·11-s + 1.42i·13-s + 0.258·15-s − 0.272·17-s + 1.17i·19-s − 1.11i·21-s + 1.06·23-s − 0.200·25-s − 0.192i·27-s − 1.53i·29-s − 1.27·31-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

\(L(1)\) \(\approx\) \(0.8124602175\)
\(L(\frac12)\) \(\approx\) \(0.8124602175\)
\(L(\frac{3}{2})\) 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 \( 1 + 5.12T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 5.12iT - 13T^{2} \)
17 \( 1 + 1.12T + 17T^{2} \)
19 \( 1 - 5.12iT - 19T^{2} \)
23 \( 1 - 5.12T + 23T^{2} \)
29 \( 1 + 8.24iT - 29T^{2} \)
31 \( 1 + 7.12T + 31T^{2} \)
37 \( 1 - 5.12iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 6.24iT - 43T^{2} \)
47 \( 1 + 13.1T + 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 6.24iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 + 4.87T + 79T^{2} \)
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

−8.611657452445501379285102495944, −7.72617962088933232204909919104, −6.52591128752080932003389042096, −6.40837591432449995517104966064, −5.44196989415582410602326255149, −4.48497369403423609453964334151, −3.66142692377574464770745891441, −3.15921978709729967992677613330, −1.91421539797541521171950899930, −0.32748077774054284701410958789, 0.76584223753163552447881125370, 2.34772436443490051628267610733, 3.08652116009663689287104432146, 3.59585452265659553413675680113, 4.98754107789656766357221443270, 5.74630505716381588764556589424, 6.52808577270932997410654582299, 7.14523424003820052189444408100, 7.45640552349936458603294612874, 8.790038710795134119865734419046

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