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Properties

Label 2-1944-216.187-c0-0-2
Degree $2$
Conductor $1944$
Sign $-0.549 + 0.835i$
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  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.5 + 0.866i)8-s + (0.266 + 0.223i)11-s + (0.766 + 0.642i)16-s + (−0.766 − 1.32i)17-s + (0.939 − 1.62i)19-s + (0.266 − 0.223i)22-s + (0.173 − 0.984i)25-s + (0.766 − 0.642i)32-s + (−1.43 + 0.524i)34-s + (−1.43 − 1.20i)38-s + (−0.326 − 1.85i)41-s + (1.17 + 0.984i)43-s + (−0.173 − 0.300i)44-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.5 + 0.866i)8-s + (0.266 + 0.223i)11-s + (0.766 + 0.642i)16-s + (−0.766 − 1.32i)17-s + (0.939 − 1.62i)19-s + (0.266 − 0.223i)22-s + (0.173 − 0.984i)25-s + (0.766 − 0.642i)32-s + (−1.43 + 0.524i)34-s + (−1.43 − 1.20i)38-s + (−0.326 − 1.85i)41-s + (1.17 + 0.984i)43-s + (−0.173 − 0.300i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.549 + 0.835i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1675, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.549 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.060256213\)
\(L(\frac12)\) \(\approx\) \(1.060256213\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
13 \( 1 + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
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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

−9.123656775490848508203634658841, −8.789092457725568352584363420194, −7.52139725599045215840140363883, −6.80860218531605099347274851585, −5.65806903935628871001916352157, −4.82304306845759910727165732387, −4.19328651141861537719199733377, −2.96713504216340982369358047273, −2.32175171856978797643088917230, −0.799154903726255189152722892458, 1.52292273774040810470822398476, 3.25609525346659460459144805620, 3.97233515403431741698803097371, 4.91111447577254248250570847255, 5.90931084040443193180121376557, 6.29082843062096928227593029663, 7.39070780356615673884179010871, 7.942679274941418946825544311499, 8.744489361128508081182812656726, 9.414756160667483969940757870972

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