Properties

Label 2-882-441.151-c1-0-0
Degree $2$
Conductor $882$
Sign $-0.994 + 0.103i$
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  + (0.900 − 0.433i)2-s + (0.184 + 1.72i)3-s + (0.623 − 0.781i)4-s + (−0.994 − 0.922i)5-s + (0.913 + 1.47i)6-s + (−2.60 + 0.464i)7-s + (0.222 − 0.974i)8-s + (−2.93 + 0.636i)9-s + (−1.29 − 0.399i)10-s + (−1.49 − 1.02i)11-s + (1.46 + 0.929i)12-s + (−3.81 − 2.60i)13-s + (−2.14 + 1.54i)14-s + (1.40 − 1.88i)15-s + (−0.222 − 0.974i)16-s + (−5.02 + 0.757i)17-s + ⋯
L(s)  = 1  + (0.637 − 0.306i)2-s + (0.106 + 0.994i)3-s + (0.311 − 0.390i)4-s + (−0.444 − 0.412i)5-s + (0.373 + 0.600i)6-s + (−0.984 + 0.175i)7-s + (0.0786 − 0.344i)8-s + (−0.977 + 0.212i)9-s + (−0.409 − 0.126i)10-s + (−0.452 − 0.308i)11-s + (0.421 + 0.268i)12-s + (−1.05 − 0.722i)13-s + (−0.573 + 0.413i)14-s + (0.362 − 0.486i)15-s + (−0.0556 − 0.243i)16-s + (−1.21 + 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.994 + 0.103i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00200200 - 0.0384777i\)
\(L(\frac12)\) \(\approx\) \(0.00200200 - 0.0384777i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 + 0.433i)T \)
3 \( 1 + (-0.184 - 1.72i)T \)
7 \( 1 + (2.60 - 0.464i)T \)
good5 \( 1 + (0.994 + 0.922i)T + (0.373 + 4.98i)T^{2} \)
11 \( 1 + (1.49 + 1.02i)T + (4.01 + 10.2i)T^{2} \)
13 \( 1 + (3.81 + 2.60i)T + (4.74 + 12.1i)T^{2} \)
17 \( 1 + (5.02 - 0.757i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-2.92 - 5.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.58 - 6.58i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (5.39 - 0.813i)T + (27.7 - 8.54i)T^{2} \)
31 \( 1 + 11.0T + 31T^{2} \)
37 \( 1 + (-2.57 + 6.55i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (-2.20 + 0.679i)T + (33.8 - 23.0i)T^{2} \)
43 \( 1 + (-6.25 - 1.92i)T + (35.5 + 24.2i)T^{2} \)
47 \( 1 + (0.0242 - 0.0116i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.232 - 0.593i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (2.34 + 10.2i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-3.46 - 4.33i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 - 5.33T + 67T^{2} \)
71 \( 1 + (4.57 - 5.74i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-5.82 + 3.97i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + 0.719T + 79T^{2} \)
83 \( 1 + (9.79 - 6.67i)T + (30.3 - 77.2i)T^{2} \)
89 \( 1 + (-0.632 - 8.44i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (7.73 - 13.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.71104509541689460953229124603, −9.605462065756996153321661567163, −9.324341964209963739982563959021, −8.076366485984180994253360765289, −7.17382516053391045805750432389, −5.69057444883625396572202805479, −5.38777524094990085312863188134, −4.10749547805369614656363504836, −3.46733367253554013397783099138, −2.42475492055995993556047551043, 0.01321191959083826199064021356, 2.29999951880386862111873671431, 3.02940635403540870436252158725, 4.29153596623946724678685201786, 5.37829677281823426792746989244, 6.52021631513149559588326457859, 7.17631671759062840660746622095, 7.41410277587110484159870034043, 8.840449361800728794312254312236, 9.478509599297440215523619869408

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