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Properties

Label 2-1944-216.115-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.5424558341\)
\(L(\frac12)\) \(\approx\) \(0.5424558341\)
\(L(1)\) 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

−9.711818846769457828272099524144, −8.854380656659749639435392447360, −8.007767051995559424343068944682, −7.39015644218788750748469970705, −6.72706687674793844627524634458, −5.79981402619027243196951588027, −4.94565169933415971208404761297, −3.89748154702166409664975750356, −2.39545538881496084653710512343, −1.66181791972155906343863642944, 0.51371093473590003338310190372, 2.04461484772460385476743235488, 3.04904879953784496565247383891, 3.84805081366725728563999595556, 5.17244680770768526418227816650, 6.24539340611664212957495523062, 6.74598755599756345168465893866, 7.933550428296403163442091963917, 8.396528963671559715430809767615, 9.017619597018793268093159431025

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