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Properties

Label 2-882-441.131-c1-0-14
Degree $2$
Conductor $882$
Sign $0.900 + 0.434i$
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  + (−0.781 − 0.623i)2-s + (−0.489 − 1.66i)3-s + (0.222 + 0.974i)4-s + (−0.0653 − 0.871i)5-s + (−0.653 + 1.60i)6-s + (0.168 + 2.64i)7-s + (0.433 − 0.900i)8-s + (−2.52 + 1.62i)9-s + (−0.492 + 0.722i)10-s + (−3.71 + 1.45i)11-s + (1.51 − 0.846i)12-s + (−0.555 + 0.217i)13-s + (1.51 − 2.16i)14-s + (−1.41 + 0.535i)15-s + (−0.900 + 0.433i)16-s + (6.68 − 2.06i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (−0.282 − 0.959i)3-s + (0.111 + 0.487i)4-s + (−0.0292 − 0.389i)5-s + (−0.266 + 0.654i)6-s + (0.0637 + 0.997i)7-s + (0.153 − 0.318i)8-s + (−0.840 + 0.542i)9-s + (−0.155 + 0.228i)10-s + (−1.12 + 0.439i)11-s + (0.436 − 0.244i)12-s + (−0.154 + 0.0604i)13-s + (0.404 − 0.579i)14-s + (−0.365 + 0.138i)15-s + (−0.225 + 0.108i)16-s + (1.62 − 0.499i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.900 + 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.920223 - 0.210543i\)
\(L(\frac12)\) \(\approx\) \(0.920223 - 0.210543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 + 0.623i)T \)
3 \( 1 + (0.489 + 1.66i)T \)
7 \( 1 + (-0.168 - 2.64i)T \)
good5 \( 1 + (0.0653 + 0.871i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (3.71 - 1.45i)T + (8.06 - 7.48i)T^{2} \)
13 \( 1 + (0.555 - 0.217i)T + (9.52 - 8.84i)T^{2} \)
17 \( 1 + (-6.68 + 2.06i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-4.36 - 2.51i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.597 + 0.643i)T + (-1.71 + 22.9i)T^{2} \)
29 \( 1 + (0.0303 + 0.0984i)T + (-23.9 + 16.3i)T^{2} \)
31 \( 1 - 7.92iT - 31T^{2} \)
37 \( 1 + (5.17 + 4.80i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-0.375 + 0.255i)T + (14.9 - 38.1i)T^{2} \)
43 \( 1 + (-6.94 - 4.73i)T + (15.7 + 40.0i)T^{2} \)
47 \( 1 + (-1.20 + 1.51i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-3.05 - 3.29i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (-5.46 + 2.62i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (0.0450 + 0.0102i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + (-16.0 + 3.66i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.08 - 1.20i)T + (53.5 + 49.6i)T^{2} \)
79 \( 1 + 4.00T + 79T^{2} \)
83 \( 1 + (0.166 - 0.423i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (8.36 - 1.26i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (7.95 - 4.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.06983262550388312297446693819, −9.206475947679113169720547252895, −8.273830419382353173735714712948, −7.73290953857622193664853466567, −6.88417783203943064636527240139, −5.51847727003157989392824162668, −5.15366229707785074430812665068, −3.19369431369233320169257465405, −2.30500317074340028462668268513, −1.05656098312249601098627758968, 0.71921801066004086235033453201, 2.89361193093278296168456567655, 3.85618726722625294937924905188, 5.13408674472424436908795498674, 5.69091874008780513438772907083, 6.90259458398795179591216403239, 7.72254595245720575033384945390, 8.432254620876337822692461690241, 9.682327698748549562519765830710, 10.06089596594692638588548858625

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