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Properties

Label 2-1776-111.62-c0-0-0
Degree $2$
Conductor $1776$
Sign $-0.665 - 0.746i$
Analytic cond. $0.886339$
Root an. cond. $0.941456$
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  + (−0.766 − 0.642i)3-s + (−0.326 + 0.118i)7-s + (0.173 + 0.984i)9-s + (−1.93 − 0.342i)13-s + (−1.11 + 1.32i)19-s + (0.326 + 0.118i)21-s + (−0.766 + 0.642i)25-s + (0.500 − 0.866i)27-s − 0.684i·31-s + (−0.5 − 0.866i)37-s + (1.26 + 1.50i)39-s + 1.28i·43-s + (−0.673 + 0.565i)49-s + (1.70 − 0.300i)57-s + (−0.173 − 0.300i)63-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)3-s + (−0.326 + 0.118i)7-s + (0.173 + 0.984i)9-s + (−1.93 − 0.342i)13-s + (−1.11 + 1.32i)19-s + (0.326 + 0.118i)21-s + (−0.766 + 0.642i)25-s + (0.500 − 0.866i)27-s − 0.684i·31-s + (−0.5 − 0.866i)37-s + (1.26 + 1.50i)39-s + 1.28i·43-s + (−0.673 + 0.565i)49-s + (1.70 − 0.300i)57-s + (−0.173 − 0.300i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $-0.665 - 0.746i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (1505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ -0.665 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1600697232\)
\(L(\frac12)\) \(\approx\) \(0.1600697232\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (1.93 + 0.342i)T + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.939 + 0.342i)T^{2} \)
19 \( 1 + (1.11 - 1.32i)T + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + 0.684iT - T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 - 1.28iT - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + 1.53T + T^{2} \)
79 \( 1 + (-0.439 - 1.20i)T + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.989953957308734404655540487991, −9.024336954137375250404670318129, −7.78107316455192027342557763326, −7.56974778101422324621460214508, −6.49692961454745000542618594822, −5.83440329840153418789614192172, −5.03739106385236260063527956738, −4.10451143515427757629211738590, −2.69748448665779232096108559716, −1.75626727381000148952863306999, 0.12237317591486658971020997456, 2.17561486053692629938508312306, 3.31525000719881296505408188134, 4.56048587632752950899567638882, 4.84045326683478855749124804508, 5.97150571281363772428624567964, 6.79186930943812815997089189544, 7.35475081617886239107459848953, 8.641263621293497998238735134307, 9.310230217329380333119662844604

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