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Properties

Label 2-882-49.8-c1-0-11
Degree $2$
Conductor $882$
Sign $-0.0960 + 0.995i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (−2.24 + 1.08i)5-s + (−2.06 + 1.64i)7-s + (0.900 − 0.433i)8-s + (2.24 + 1.08i)10-s + (−1.55 − 1.94i)11-s + (0.986 + 1.23i)13-s + (2.57 + 0.588i)14-s + (−0.900 − 0.433i)16-s + (−0.356 − 1.56i)17-s + 2.75·19-s + (−0.554 − 2.43i)20-s + (−0.554 + 2.43i)22-s + (0.841 − 3.68i)23-s + ⋯
L(s)  = 1  + (−0.440 − 0.552i)2-s + (−0.111 + 0.487i)4-s + (−1.00 + 0.483i)5-s + (−0.781 + 0.623i)7-s + (0.318 − 0.153i)8-s + (0.710 + 0.342i)10-s + (−0.468 − 0.587i)11-s + (0.273 + 0.343i)13-s + (0.689 + 0.157i)14-s + (−0.225 − 0.108i)16-s + (−0.0865 − 0.379i)17-s + 0.631·19-s + (−0.124 − 0.543i)20-s + (−0.118 + 0.518i)22-s + (0.175 − 0.768i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0960 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0960 + 0.995i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.0960 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.402588 - 0.443294i\)
\(L(\frac12)\) \(\approx\) \(0.402588 - 0.443294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 \)
7 \( 1 + (2.06 - 1.64i)T \)
good5 \( 1 + (2.24 - 1.08i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (1.55 + 1.94i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.986 - 1.23i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.356 + 1.56i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 - 2.75T + 19T^{2} \)
23 \( 1 + (-0.841 + 3.68i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.911 + 3.99i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 - 7.03T + 31T^{2} \)
37 \( 1 + (2.18 + 9.58i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-4.40 + 2.12i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-5.37 - 2.58i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (7.71 + 9.67i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.652 - 2.86i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (-8.49 - 4.09i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (1.07 + 4.71i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + 0.295T + 67T^{2} \)
71 \( 1 + (0.484 - 2.12i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.77 + 3.47i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + (-5.60 + 7.02i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (5.92 - 7.43i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 - 0.180T + 97T^{2} \)
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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

−9.944517733603208023040780047718, −9.097938905915401698567745569742, −8.336802803199362645303106860405, −7.50555906792953093726162832523, −6.63333564252613702376836922068, −5.56887707239562666231071838003, −4.19308830603866237939807845396, −3.28123295357519941510105013803, −2.46270824244381794261319830957, −0.40648194210481450987576461414, 1.08020113940707389492009246057, 3.08847445742321912692240746400, 4.16642535793103971184828472065, 5.06181995475910074901960166383, 6.22010851617762449139509543896, 7.13884702382727992863898920702, 7.82050755284329378362659358242, 8.476829618716224131686294771708, 9.554005584950616143741115859221, 10.13918358740378237426879688979

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