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Properties

Label 2-1944-9.4-c1-0-0
Degree $2$
Conductor $1944$
Sign $-0.766 - 0.642i$
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  + (0.766 − 1.32i)5-s + (1.26 + 2.19i)7-s + (−2.20 − 3.82i)11-s + (−1.97 + 3.41i)13-s − 3.69·17-s − 7.10·19-s + (−2.35 + 4.08i)23-s + (1.32 + 2.29i)25-s + (1.93 + 3.35i)29-s + (−1.64 + 2.84i)31-s + 3.87·35-s − 7.92·37-s + (−2.66 + 4.61i)41-s + (−0.868 − 1.50i)43-s + (−0.233 − 0.405i)47-s + ⋯
L(s)  = 1  + (0.342 − 0.593i)5-s + (0.478 + 0.828i)7-s + (−0.665 − 1.15i)11-s + (−0.546 + 0.947i)13-s − 0.896·17-s − 1.63·19-s + (−0.491 + 0.851i)23-s + (0.265 + 0.459i)25-s + (0.360 + 0.623i)29-s + (−0.295 + 0.511i)31-s + 0.655·35-s − 1.30·37-s + (−0.416 + 0.721i)41-s + (−0.132 − 0.229i)43-s + (−0.0341 − 0.0591i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.766 - 0.642i$
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: \((2,\ 1944,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5507179370\)
\(L(\frac12)\) \(\approx\) \(0.5507179370\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.766 + 1.32i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.26 - 2.19i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.20 + 3.82i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.97 - 3.41i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.69T + 17T^{2} \)
19 \( 1 + 7.10T + 19T^{2} \)
23 \( 1 + (2.35 - 4.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.93 - 3.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.64 - 2.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.92T + 37T^{2} \)
41 \( 1 + (2.66 - 4.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.868 + 1.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.233 + 0.405i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.80T + 53T^{2} \)
59 \( 1 + (-4.02 + 6.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.24 - 5.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.28 + 9.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.46T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + (-3.60 - 6.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.32 + 9.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.41T + 89T^{2} \)
97 \( 1 + (-7.62 - 13.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.184709833242847716749427122283, −8.725312256844810622085990144960, −8.261758711732171974292242290053, −7.06361161877643195287216274799, −6.26423851019741298800323594226, −5.35229566980911143027409619033, −4.85530407507148697795708101078, −3.72116924263603208900812171436, −2.45649284784585577552815802138, −1.66509045846723258872204619763, 0.17888126263532714268285114198, 2.03172390724972115081817433287, 2.63586242330817794443513004181, 4.13444370144164908283290534625, 4.63431144548745535630201041477, 5.69480922692478269307747101510, 6.72060663917133295118971178759, 7.20082365360389962352091297629, 8.083403724379967652071378210805, 8.745707266691548477111723850626

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