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Properties

Label 2-1776-1776.221-c0-0-11
Degree $2$
Conductor $1776$
Sign $-0.382 + 0.923i$
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  + (0.980 − 0.195i)2-s + (−0.707 − 0.707i)3-s + (0.923 − 0.382i)4-s + (0.275 − 0.275i)5-s + (−0.831 − 0.555i)6-s − 1.84i·7-s + (0.831 − 0.555i)8-s + 1.00i·9-s + (0.216 − 0.324i)10-s + (−0.923 − 0.382i)12-s + (−0.360 − 1.81i)14-s − 0.390·15-s + (0.707 − 0.707i)16-s − 1.96·17-s + (0.195 + 0.980i)18-s + ⋯
L(s)  = 1  + (0.980 − 0.195i)2-s + (−0.707 − 0.707i)3-s + (0.923 − 0.382i)4-s + (0.275 − 0.275i)5-s + (−0.831 − 0.555i)6-s − 1.84i·7-s + (0.831 − 0.555i)8-s + 1.00i·9-s + (0.216 − 0.324i)10-s + (−0.923 − 0.382i)12-s + (−0.360 − 1.81i)14-s − 0.390·15-s + (0.707 − 0.707i)16-s − 1.96·17-s + (0.195 + 0.980i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.690623078\)
\(L(\frac12)\) \(\approx\) \(1.690623078\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.980 + 0.195i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + (-0.275 + 0.275i)T - iT^{2} \)
7 \( 1 + 1.84iT - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + 1.96T + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - 1.11iT - T^{2} \)
29 \( 1 + (-1.17 - 1.17i)T + iT^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-1.17 + 1.17i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 0.765iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - 1.66iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.451118899139183416818445848367, −8.125896627630244212639877747225, −7.13855354213905767185864206177, −6.92194522147752439482631382831, −6.04397563205897181424399385002, −4.99829081561921484987032694707, −4.45228505974117398591719184240, −3.46474460425601348118507396942, −2.04078856541315556191387271725, −1.05272713732189888339545446342, 2.30427234832396936626627131555, 2.80592922572150122596789299663, 4.30598470229856745786715453810, 4.74921722150574917874063990691, 5.78001483824260009325182788908, 6.25277424323478264922484207004, 6.79630166424648806308649188743, 8.368632438285008502673136224548, 8.809076472913822295908212264196, 9.880130108447431750809984972263

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