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Properties

Label 2-1944-216.115-c0-0-3
Degree $2$
Conductor $1944$
Sign $0.448 + 0.893i$
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  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.500 − 0.866i)8-s + (0.326 − 1.85i)11-s + (0.173 − 0.984i)16-s + (0.173 + 0.300i)17-s + (−0.766 + 1.32i)19-s + (−0.326 − 1.85i)22-s + (−0.939 + 0.342i)25-s + (−0.173 − 0.984i)32-s + (0.266 + 0.223i)34-s + (−0.266 + 1.50i)38-s + (1.43 + 0.524i)41-s + (0.0603 − 0.342i)43-s + (−0.939 − 1.62i)44-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.500 − 0.866i)8-s + (0.326 − 1.85i)11-s + (0.173 − 0.984i)16-s + (0.173 + 0.300i)17-s + (−0.766 + 1.32i)19-s + (−0.326 − 1.85i)22-s + (−0.939 + 0.342i)25-s + (−0.173 − 0.984i)32-s + (0.266 + 0.223i)34-s + (−0.266 + 1.50i)38-s + (1.43 + 0.524i)41-s + (0.0603 − 0.342i)43-s + (−0.939 − 1.62i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.448 + 0.893i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.448 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.054964160\)
\(L(\frac12)\) \(\approx\) \(2.054964160\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 \)
good5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.173 - 0.984i)T^{2} \)
11 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
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 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
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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

−9.372618052539984139178020586916, −8.352898261070544759897713882656, −7.70795863204811406415753424236, −6.46207025537050598265880190773, −5.98886691402534161226146891716, −5.32625333464810837634861484771, −4.02458419632516746309826095225, −3.59309863149272016895102466001, −2.49459858851301469656874134790, −1.24007254212845196666883464010, 1.92458444129011073624186614213, 2.71074827748417174778371657346, 4.07348333411546764142899657578, 4.52645776261526215737395790570, 5.41255243439090476361922384949, 6.39051895939477793237539236336, 7.10461687559508584111483505899, 7.60096088097759777464682341406, 8.646792057957175114646060873351, 9.518615950022094970737417673466

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