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Properties

Label 2-1944-216.211-c0-0-1
Degree $2$
Conductor $1944$
Sign $0.230 + 0.973i$
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  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (0.500 + 0.866i)8-s + (−1.76 − 0.642i)11-s + (−0.939 − 0.342i)16-s + (0.173 − 0.300i)17-s + (−0.766 − 1.32i)19-s + (1.76 − 0.642i)22-s + (0.766 − 0.642i)25-s + (0.939 − 0.342i)32-s + (0.0603 + 0.342i)34-s + (1.43 + 0.524i)38-s + (−1.17 − 0.984i)41-s + (−0.326 − 0.118i)43-s + (−0.939 + 1.62i)44-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (0.500 + 0.866i)8-s + (−1.76 − 0.642i)11-s + (−0.939 − 0.342i)16-s + (0.173 − 0.300i)17-s + (−0.766 − 1.32i)19-s + (1.76 − 0.642i)22-s + (0.766 − 0.642i)25-s + (0.939 − 0.342i)32-s + (0.0603 + 0.342i)34-s + (1.43 + 0.524i)38-s + (−1.17 − 0.984i)41-s + (−0.326 − 0.118i)43-s + (−0.939 + 1.62i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.230 + 0.973i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4596812620\)
\(L(\frac12)\) \(\approx\) \(0.4596812620\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 \)
good5 \( 1 + (-0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.939 - 0.342i)T^{2} \)
11 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.939 + 0.342i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
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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

−8.968756376056695800100420549335, −8.450222950054644083331716651420, −7.73584405098279159471646937650, −6.95929536155660459627817113387, −6.18479983637114234351524571687, −5.22868804115200494904214797532, −4.72843888119058415264620621686, −3.06610146756619313699446218112, −2.15705915063208319161240804429, −0.41315284090464139611703878691, 1.58901617012545022043932721233, 2.56481035797116139691459758888, 3.47863453984749721220659713187, 4.55544866675015027259945620086, 5.46746549468543278305777352689, 6.61824920494818026106827316759, 7.48378815973597625807639338496, 8.125852172633745048437240182009, 8.666755553435167629706648891894, 9.816403528501894126008669626834

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