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Properties

Label 2-882-441.16-c1-0-27
Degree $2$
Conductor $882$
Sign $0.603 - 0.797i$
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  + (−0.988 − 0.149i)2-s + (1.16 + 1.27i)3-s + (0.955 + 0.294i)4-s + (1.30 + 0.626i)5-s + (−0.966 − 1.43i)6-s + (1.76 + 1.96i)7-s + (−0.900 − 0.433i)8-s + (−0.264 + 2.98i)9-s + (−1.19 − 0.813i)10-s + (3.11 − 3.90i)11-s + (0.740 + 1.56i)12-s + (−1.86 − 0.281i)13-s + (−1.45 − 2.20i)14-s + (0.721 + 2.39i)15-s + (0.826 + 0.563i)16-s + (7.50 − 2.31i)17-s + ⋯
L(s)  = 1  + (−0.699 − 0.105i)2-s + (0.675 + 0.737i)3-s + (0.477 + 0.147i)4-s + (0.582 + 0.280i)5-s + (−0.394 − 0.586i)6-s + (0.668 + 0.744i)7-s + (−0.318 − 0.153i)8-s + (−0.0881 + 0.996i)9-s + (−0.377 − 0.257i)10-s + (0.938 − 1.17i)11-s + (0.213 + 0.451i)12-s + (−0.518 − 0.0781i)13-s + (−0.388 − 0.590i)14-s + (0.186 + 0.618i)15-s + (0.206 + 0.140i)16-s + (1.82 − 0.561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.603 - 0.797i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.603 - 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62952 + 0.810185i\)
\(L(\frac12)\) \(\approx\) \(1.62952 + 0.810185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.988 + 0.149i)T \)
3 \( 1 + (-1.16 - 1.27i)T \)
7 \( 1 + (-1.76 - 1.96i)T \)
good5 \( 1 + (-1.30 - 0.626i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-3.11 + 3.90i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.86 + 0.281i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (-7.50 + 2.31i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-1.48 + 2.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.967 - 4.23i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-2.74 - 2.54i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (2.63 - 4.56i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.71 + 5.29i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-0.265 - 3.53i)T + (-40.5 + 6.11i)T^{2} \)
43 \( 1 + (-0.802 + 10.7i)T + (-42.5 - 6.40i)T^{2} \)
47 \( 1 + (5.34 + 0.806i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (6.28 - 5.83i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (-1.04 + 13.9i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (6.50 - 2.00i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (5.02 - 8.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.93 - 12.8i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (1.75 - 4.47i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (0.119 + 0.206i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.40 - 0.664i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (-17.8 + 2.69i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (1.24 - 2.15i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.04356242742890055695323371025, −9.369222590342178510270723208825, −8.781505390568266217062091184520, −8.003282109464301768200711315026, −7.09367982625823847182356329353, −5.74588442662299828709960736987, −5.14175738805439123838181364466, −3.53842139419135865214578387062, −2.79283020674773655731594130892, −1.51366104297258711403782057251, 1.26597030645468128891667887962, 1.87042695230447333120953438307, 3.39708488953787546901044770925, 4.64118717034131573080295082737, 5.95014161159570180570363961625, 6.84556496744276461835398234317, 7.70361042335394312709802518971, 8.080473762319205245404987341524, 9.318545678658243432382197079451, 9.743933808377409543198471343324

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