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Properties

Label 2-1944-24.5-c0-0-4
Degree $2$
Conductor $1944$
Sign $0.707 - 0.707i$
Analytic cond. $0.970182$
Root an. cond. $0.984978$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + 1.41·5-s + 7-s + (0.707 + 0.707i)8-s + (−1.00 + 1.00i)10-s − i·13-s + (−0.707 + 0.707i)14-s − 1.00·16-s + 1.41i·17-s + i·19-s − 1.41i·20-s + 1.41i·23-s + 1.00·25-s + (0.707 + 0.707i)26-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + 1.41·5-s + 7-s + (0.707 + 0.707i)8-s + (−1.00 + 1.00i)10-s − i·13-s + (−0.707 + 0.707i)14-s − 1.00·16-s + 1.41i·17-s + i·19-s − 1.41i·20-s + 1.41i·23-s + 1.00·25-s + (0.707 + 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.156176915\)
\(L(\frac12)\) \(\approx\) \(1.156176915\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
good5 \( 1 - 1.41T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.334528501111118880073586868058, −8.724037502932333006405820086962, −7.88797379362825559956807988682, −7.33823019209642055567371554783, −6.07737540406506055558584639307, −5.69870925495695370086835407673, −5.11069575605308624946143039482, −3.73399661856818561901693396929, −2.02049087937541360844441506960, −1.52703396807628706455572392143, 1.30176333535926952156344348775, 2.19604389965158858972674501915, 2.92998011807298066742639966130, 4.54929799002007938829681284848, 4.94437388266083209238158860047, 6.31533481245987775461089904822, 6.96958125679090726873658485625, 7.912944948005591014559206204237, 8.798971289930732727145472742855, 9.364615951608641700611571304742

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