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Properties

Label 2-882-441.16-c1-0-26
Degree $2$
Conductor $882$
Sign $0.856 - 0.515i$
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 + (−1.68 + 0.385i)3-s + (0.955 + 0.294i)4-s + (1.02 + 0.492i)5-s + (−1.72 + 0.129i)6-s + (2.30 + 1.29i)7-s + (0.900 + 0.433i)8-s + (2.70 − 1.30i)9-s + (0.938 + 0.639i)10-s + (2.68 − 3.37i)11-s + (−1.72 − 0.129i)12-s + (−2.42 − 0.365i)13-s + (2.08 + 1.62i)14-s + (−1.91 − 0.437i)15-s + (0.826 + 0.563i)16-s + (0.333 − 0.102i)17-s + ⋯
L(s)  = 1  + (0.699 + 0.105i)2-s + (−0.974 + 0.222i)3-s + (0.477 + 0.147i)4-s + (0.457 + 0.220i)5-s + (−0.705 + 0.0528i)6-s + (0.871 + 0.490i)7-s + (0.318 + 0.153i)8-s + (0.900 − 0.433i)9-s + (0.296 + 0.202i)10-s + (0.810 − 1.01i)11-s + (−0.498 − 0.0373i)12-s + (−0.672 − 0.101i)13-s + (0.557 + 0.434i)14-s + (−0.495 − 0.113i)15-s + (0.206 + 0.140i)16-s + (0.0808 − 0.0249i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.856 - 0.515i$
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: \((2,\ 882,\ (\ :1/2),\ 0.856 - 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12042 + 0.589271i\)
\(L(\frac12)\) \(\approx\) \(2.12042 + 0.589271i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 0.385i)T \)
7 \( 1 + (-2.30 - 1.29i)T \)
good5 \( 1 + (-1.02 - 0.492i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-2.68 + 3.37i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (2.42 + 0.365i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (-0.333 + 0.102i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-0.320 + 0.554i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.692 - 3.03i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-6.31 - 5.86i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (-0.746 + 1.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.88 - 7.31i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (0.0918 + 1.22i)T + (-40.5 + 6.11i)T^{2} \)
43 \( 1 + (-0.720 + 9.61i)T + (-42.5 - 6.40i)T^{2} \)
47 \( 1 + (-1.40 - 0.212i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (7.82 - 7.25i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (0.905 - 12.0i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-8.97 + 2.76i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (0.320 - 0.555i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.80 + 12.2i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.871 + 2.22i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (4.43 + 7.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.9 - 1.80i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (9.30 - 1.40i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (8.34 - 14.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.46748068373122489744447644291, −9.513827322665306564910868737959, −8.509206881929621959352057415233, −7.40573307810083365677239372622, −6.43280488447681925107336527088, −5.78720138805026333056563233712, −5.02578977731850648379033480639, −4.15573384809782441185374813732, −2.82568800949506064141069093336, −1.34100199705392392805313388096, 1.22298445710671000696851366585, 2.25964754965247741721202362714, 4.20660901952527211757405963244, 4.65778236910888640816101482185, 5.57831634700619600516777448749, 6.52559879718094161738493997208, 7.23609896714633907315074722998, 8.111419124947825815673440103402, 9.681789489321417193110524995050, 10.05125206300741861923363115190

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