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Properties

Label 2-1944-9.4-c1-0-29
Degree $2$
Conductor $1944$
Sign $-0.173 + 0.984i$
Analytic cond. $15.5229$
Root an. cond. $3.93991$
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  + (0.939 − 1.62i)5-s + (−0.439 − 0.761i)7-s + (−0.613 − 1.06i)11-s + (2.55 − 4.42i)13-s + 6.06·17-s − 3.83·19-s + (−1.64 + 2.84i)23-s + (0.733 + 1.27i)25-s + (−0.826 − 1.43i)29-s + (2.28 − 3.96i)31-s − 1.65·35-s − 9.24·37-s + (0.511 − 0.885i)41-s + (−3.83 − 6.63i)43-s + (1.93 + 3.35i)47-s + ⋯
L(s)  = 1  + (0.420 − 0.727i)5-s + (−0.166 − 0.287i)7-s + (−0.184 − 0.320i)11-s + (0.708 − 1.22i)13-s + 1.47·17-s − 0.880·19-s + (−0.343 + 0.594i)23-s + (0.146 + 0.254i)25-s + (−0.153 − 0.265i)29-s + (0.410 − 0.711i)31-s − 0.279·35-s − 1.52·37-s + (0.0798 − 0.138i)41-s + (−0.584 − 1.01i)43-s + (0.282 + 0.490i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(15.5229\)
Root analytic conductor: \(3.93991\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.661008346\)
\(L(\frac12)\) \(\approx\) \(1.661008346\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.939 + 1.62i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.439 + 0.761i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.613 + 1.06i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.55 + 4.42i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.06T + 17T^{2} \)
19 \( 1 + 3.83T + 19T^{2} \)
23 \( 1 + (1.64 - 2.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.826 + 1.43i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.28 + 3.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.24T + 37T^{2} \)
41 \( 1 + (-0.511 + 0.885i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.83 + 6.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.93 - 3.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.90T + 53T^{2} \)
59 \( 1 + (-2.13 + 3.69i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.237 + 0.411i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.35 + 9.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 4.68T + 73T^{2} \)
79 \( 1 + (3.88 + 6.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.95 - 15.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + (6.01 + 10.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.910742621652761514180494909755, −8.199287984051373300796122406581, −7.60825137148998538670043805396, −6.47065972293931037851624736853, −5.60478217236928183521856449079, −5.18644111133241998422916909639, −3.87327577235197191543393098866, −3.16763140533063905078508976757, −1.74501040200199628086014825078, −0.61575327861448838687141969137, 1.50721854241807753178441982535, 2.53951957528363474800658671607, 3.49593693349935609814736272317, 4.48060413309100986455718078476, 5.51007497739320323037796896348, 6.41641942549543212620365517956, 6.80258107069607051538497300564, 7.86987531073204006851139294809, 8.686984320356643793036465767814, 9.389936371084852873374891092975

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