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Properties

Label 2-1944-1944.187-c0-0-0
Degree $2$
Conductor $1944$
Sign $0.167 + 0.985i$
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  + (0.987 − 0.154i)2-s + (0.249 − 0.968i)3-s + (0.952 − 0.305i)4-s + (0.0968 − 0.995i)6-s + (0.893 − 0.448i)8-s + (−0.875 − 0.483i)9-s + (−0.422 − 0.0328i)11-s + (−0.0581 − 0.998i)12-s + (0.813 − 0.581i)16-s + (0.256 + 0.594i)17-s + (−0.939 − 0.342i)18-s + (−0.515 − 0.692i)19-s + (−0.422 + 0.0328i)22-s + (−0.211 − 0.977i)24-s + (0.657 − 0.753i)25-s + ⋯
L(s)  = 1  + (0.987 − 0.154i)2-s + (0.249 − 0.968i)3-s + (0.952 − 0.305i)4-s + (0.0968 − 0.995i)6-s + (0.893 − 0.448i)8-s + (−0.875 − 0.483i)9-s + (−0.422 − 0.0328i)11-s + (−0.0581 − 0.998i)12-s + (0.813 − 0.581i)16-s + (0.256 + 0.594i)17-s + (−0.939 − 0.342i)18-s + (−0.515 − 0.692i)19-s + (−0.422 + 0.0328i)22-s + (−0.211 − 0.977i)24-s + (0.657 − 0.753i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.167 + 0.985i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.167 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.242978372\)
\(L(\frac12)\) \(\approx\) \(2.242978372\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.987 + 0.154i)T \)
3 \( 1 + (-0.249 + 0.968i)T \)
good5 \( 1 + (-0.657 + 0.753i)T^{2} \)
7 \( 1 + (0.963 + 0.268i)T^{2} \)
11 \( 1 + (0.422 + 0.0328i)T + (0.987 + 0.154i)T^{2} \)
13 \( 1 + (0.790 + 0.612i)T^{2} \)
17 \( 1 + (-0.256 - 0.594i)T + (-0.686 + 0.727i)T^{2} \)
19 \( 1 + (0.515 + 0.692i)T + (-0.286 + 0.957i)T^{2} \)
23 \( 1 + (-0.249 + 0.968i)T^{2} \)
29 \( 1 + (-0.466 - 0.884i)T^{2} \)
31 \( 1 + (-0.996 - 0.0774i)T^{2} \)
37 \( 1 + (0.0581 - 0.998i)T^{2} \)
41 \( 1 + (-0.693 - 1.79i)T + (-0.740 + 0.672i)T^{2} \)
43 \( 1 + (0.179 - 0.697i)T + (-0.875 - 0.483i)T^{2} \)
47 \( 1 + (-0.996 + 0.0774i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.701 - 1.46i)T + (-0.627 - 0.778i)T^{2} \)
61 \( 1 + (-0.813 - 0.581i)T^{2} \)
67 \( 1 + (0.231 - 0.139i)T + (0.466 - 0.884i)T^{2} \)
71 \( 1 + (0.993 - 0.116i)T^{2} \)
73 \( 1 + (0.0324 + 0.0213i)T + (0.396 + 0.918i)T^{2} \)
79 \( 1 + (0.211 - 0.977i)T^{2} \)
83 \( 1 + (-0.206 + 0.535i)T + (-0.740 - 0.672i)T^{2} \)
89 \( 1 + (0.0694 - 1.19i)T + (-0.993 - 0.116i)T^{2} \)
97 \( 1 + (-1.81 - 0.826i)T + (0.657 + 0.753i)T^{2} \)
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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

−9.105773989803711070744031997730, −8.136909689933823856668288071976, −7.58347634318242178683169660196, −6.56915665928016545285056321149, −6.20367598359605995647524719809, −5.17474013091481666827157158864, −4.29027805338964791187115606901, −3.11777871300194509514752411949, −2.45684714333189824178288920073, −1.29466240721205576840129234541, 2.05649064077215304797758002234, 3.08664233344402084525026117329, 3.78492584876653728153605199075, 4.72209687790093513026358460063, 5.34225727395875611619775553327, 6.10667271533242551145872103261, 7.16914172441971291215102378606, 7.920578356247364065868406055255, 8.744335878736797950263695168316, 9.627830625356742312590374749592

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