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Properties

Label 2-876-219.194-c0-0-0
Degree $2$
Conductor $876$
Sign $0.999 + 0.0319i$
Analytic cond. $0.437180$
Root an. cond. $0.661196$
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  + (0.866 + 0.5i)3-s + (−0.515 − 1.92i)7-s + (0.499 + 0.866i)9-s + (1.28 + 0.597i)13-s + (−0.223 + 0.266i)19-s + (0.515 − 1.92i)21-s + (−0.984 − 0.173i)25-s + 0.999i·27-s + (1.48 − 1.03i)31-s + (−1.32 + 1.11i)37-s + (0.811 + 1.15i)39-s + (−1.58 + 0.424i)43-s + (−2.57 + 1.48i)49-s + (−0.326 + 0.118i)57-s + (−0.233 + 0.642i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.515 − 1.92i)7-s + (0.499 + 0.866i)9-s + (1.28 + 0.597i)13-s + (−0.223 + 0.266i)19-s + (0.515 − 1.92i)21-s + (−0.984 − 0.173i)25-s + 0.999i·27-s + (1.48 − 1.03i)31-s + (−1.32 + 1.11i)37-s + (0.811 + 1.15i)39-s + (−1.58 + 0.424i)43-s + (−2.57 + 1.48i)49-s + (−0.326 + 0.118i)57-s + (−0.233 + 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(876\)    =    \(2^{2} \cdot 3 \cdot 73\)
Sign: $0.999 + 0.0319i$
Analytic conductor: \(0.437180\)
Root analytic conductor: \(0.661196\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{876} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 876,\ (\ :0),\ 0.999 + 0.0319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.284335824\)
\(L(\frac12)\) \(\approx\) \(1.284335824\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + iT \)
good5 \( 1 + (0.984 + 0.173i)T^{2} \)
7 \( 1 + (0.515 + 1.92i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.984 - 0.173i)T^{2} \)
13 \( 1 + (-1.28 - 0.597i)T + (0.642 + 0.766i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.223 - 0.266i)T + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (0.984 - 0.173i)T^{2} \)
31 \( 1 + (-1.48 + 1.03i)T + (0.342 - 0.939i)T^{2} \)
37 \( 1 + (1.32 - 1.11i)T + (0.173 - 0.984i)T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (1.58 - 0.424i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.642 - 0.766i)T^{2} \)
53 \( 1 + (-0.984 + 0.173i)T^{2} \)
59 \( 1 + (0.642 + 0.766i)T^{2} \)
61 \( 1 + (0.233 - 0.642i)T + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.439 - 1.20i)T + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
79 \( 1 + (0.642 + 1.76i)T + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.766 + 0.642i)T^{2} \)
97 \( 1 + (0.592 - 0.342i)T + (0.5 - 0.866i)T^{2} \)
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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

−10.14847663027471923878196318696, −9.754998969053043616938928918240, −8.582975870235400204792974471333, −7.934453507524814287270604969800, −7.00816834190973115282769354381, −6.22362363066231412416589952140, −4.59266172651457502433097959950, −3.95020842712404941683038229021, −3.22237672484143143401125251078, −1.52549644750459409713258768050, 1.80949748560620807042071841489, 2.85713821964096419438579830728, 3.64212797802653477319939507111, 5.26487420300633357771900571856, 6.12167348303368771039523245804, 6.83116252196120273159809240445, 8.249826599457987182356055062699, 8.515242543502443918124093847046, 9.301957642527141087249358528858, 10.11343249343541462199269267416

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