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Properties

Label 2-3872-8.5-c1-0-64
Degree $2$
Conductor $3872$
Sign $0.962 + 0.270i$
Analytic cond. $30.9180$
Root an. cond. $5.56040$
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  + 1.47i·3-s − 3.55i·5-s + 3.85·7-s + 0.828·9-s − 4.77i·13-s + 5.24·15-s + 1.59·17-s + 4.14i·19-s + 5.67i·21-s + 3.24·23-s − 7.65·25-s + 5.64i·27-s + 7.39i·29-s + 3.58·31-s − 13.6i·35-s + ⋯
L(s)  = 1  + 0.850i·3-s − 1.59i·5-s + 1.45·7-s + 0.276·9-s − 1.32i·13-s + 1.35·15-s + 0.386·17-s + 0.950i·19-s + 1.23i·21-s + 0.676·23-s − 1.53·25-s + 1.08i·27-s + 1.37i·29-s + 0.644·31-s − 2.31i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3872\)    =    \(2^{5} \cdot 11^{2}\)
Sign: $0.962 + 0.270i$
Analytic conductor: \(30.9180\)
Root analytic conductor: \(5.56040\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3872} (1937, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3872,\ (\ :1/2),\ 0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.616286775\)
\(L(\frac12)\) \(\approx\) \(2.616286775\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 - 1.47iT - 3T^{2} \)
5 \( 1 + 3.55iT - 5T^{2} \)
7 \( 1 - 3.85T + 7T^{2} \)
13 \( 1 + 4.77iT - 13T^{2} \)
17 \( 1 - 1.59T + 17T^{2} \)
19 \( 1 - 4.14iT - 19T^{2} \)
23 \( 1 - 3.24T + 23T^{2} \)
29 \( 1 - 7.39iT - 29T^{2} \)
31 \( 1 - 3.58T + 31T^{2} \)
37 \( 1 - 0.610iT - 37T^{2} \)
41 \( 1 - 9.29T + 41T^{2} \)
43 \( 1 - 3.06iT - 43T^{2} \)
47 \( 1 - 4.82T + 47T^{2} \)
53 \( 1 - 4.16iT - 53T^{2} \)
59 \( 1 + 5.64iT - 59T^{2} \)
61 \( 1 + 2.16iT - 61T^{2} \)
67 \( 1 - 1.47iT - 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 9.29T + 73T^{2} \)
79 \( 1 - 3.19T + 79T^{2} \)
83 \( 1 + 12.6iT - 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−8.373405701546691642087916682312, −8.024746502684347321882817765172, −7.24996938015187579315544456116, −5.72447447181146649933100951560, −5.31672524372421251647432626733, −4.66349418915713888592576612576, −4.18008197313555093851788244708, −3.10149325880717425190164337316, −1.59129373910492473709592868180, −0.978520883527378069562212925311, 1.09583603885873146780292913858, 2.15618262981741220184344257866, 2.62048338879653998739481024043, 3.99033432761011528842977428756, 4.60234541276738879319834782372, 5.73662510139783797293324236554, 6.59075326527496090717057467888, 7.05930634067565375833014767949, 7.59465449153024373668781740341, 8.233468776525767726117249986807

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