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Properties

Label 2-1776-1776.221-c0-0-5
Degree $2$
Conductor $1776$
Sign $0.923 - 0.382i$
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  + i·2-s + (0.707 − 0.707i)3-s − 4-s + (−1 + i)5-s + (0.707 + 0.707i)6-s − i·7-s − i·8-s − 1.00i·9-s + (−1 − i)10-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + 14-s + 1.41i·15-s + 16-s + 17-s + ⋯
L(s)  = 1  + i·2-s + (0.707 − 0.707i)3-s − 4-s + (−1 + i)5-s + (0.707 + 0.707i)6-s − i·7-s − i·8-s − 1.00i·9-s + (−1 − i)10-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + 14-s + 1.41i·15-s + 16-s + 17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.923 - 0.382i$
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: \((2,\ 1776,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.156647003\)
\(L(\frac12)\) \(\approx\) \(1.156647003\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
good5 \( 1 + (1 - i)T - iT^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
59 \( 1 + (-1 + i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 - iT - T^{2} \)
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

−9.360361591165990799322227165982, −8.252845200728925715451259147388, −7.80049876604908852406552877459, −7.32679850238059760365951089402, −6.55431276306578556216345848095, −5.96626695592450354259384919551, −4.15834993489228568097480162240, −3.95533944100844828531669059399, −2.88409717561937971908460859770, −1.02205453255516235267879188464, 1.31628601342381893272028984526, 2.61656112252936781010508748306, 3.68015090966488640776797502712, 4.08959093289878904249730327305, 5.07184312468271692061769994372, 5.79606467570817623557131077184, 7.44624229228111263284162194550, 8.547385493966478068082054653004, 8.686379503229682632802842170519, 9.193914986125483523397936238739

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