Properties

Label 2-882-441.151-c1-0-34
Degree $2$
Conductor $882$
Sign $0.543 + 0.839i$
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 + (1.31 − 1.12i)3-s + (0.623 − 0.781i)4-s + (−0.191 − 0.177i)5-s + (−0.697 + 1.58i)6-s + (2.18 − 1.48i)7-s + (−0.222 + 0.974i)8-s + (0.466 − 2.96i)9-s + (0.249 + 0.0769i)10-s + (−2.47 − 1.68i)11-s + (−0.0589 − 1.73i)12-s + (5.21 + 3.55i)13-s + (−1.32 + 2.28i)14-s + (−0.452 − 0.0184i)15-s + (−0.222 − 0.974i)16-s + (0.322 − 0.0486i)17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (0.760 − 0.649i)3-s + (0.311 − 0.390i)4-s + (−0.0856 − 0.0794i)5-s + (−0.284 + 0.647i)6-s + (0.826 − 0.562i)7-s + (−0.0786 + 0.344i)8-s + (0.155 − 0.987i)9-s + (0.0789 + 0.0243i)10-s + (−0.745 − 0.508i)11-s + (−0.0170 − 0.499i)12-s + (1.44 + 0.985i)13-s + (−0.354 + 0.611i)14-s + (−0.116 − 0.00475i)15-s + (−0.0556 − 0.243i)16-s + (0.0782 − 0.0117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.543 + 0.839i$
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: \((2,\ 882,\ (\ :1/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43370 - 0.779399i\)
\(L(\frac12)\) \(\approx\) \(1.43370 - 0.779399i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 + 1.12i)T \)
7 \( 1 + (-2.18 + 1.48i)T \)
good5 \( 1 + (0.191 + 0.177i)T + (0.373 + 4.98i)T^{2} \)
11 \( 1 + (2.47 + 1.68i)T + (4.01 + 10.2i)T^{2} \)
13 \( 1 + (-5.21 - 3.55i)T + (4.74 + 12.1i)T^{2} \)
17 \( 1 + (-0.322 + 0.0486i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-2.66 - 4.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.55 + 6.50i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (-8.83 + 1.33i)T + (27.7 - 8.54i)T^{2} \)
31 \( 1 + 5.86T + 31T^{2} \)
37 \( 1 + (2.37 - 6.05i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (2.29 - 0.707i)T + (33.8 - 23.0i)T^{2} \)
43 \( 1 + (-2.14 - 0.661i)T + (35.5 + 24.2i)T^{2} \)
47 \( 1 + (-0.668 + 0.321i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (3.59 + 9.16i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-1.51 - 6.65i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (3.13 + 3.92i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + (1.98 - 2.48i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-10.3 + 7.04i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 - 9.76T + 79T^{2} \)
83 \( 1 + (11.6 - 7.93i)T + (30.3 - 77.2i)T^{2} \)
89 \( 1 + (-0.359 - 4.79i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (-5.72 + 9.91i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.984516876524018234848249318207, −8.711702189915008741170124700522, −8.346599091095185986687679686379, −7.75802572929967383349928786243, −6.68877065737920543958271320826, −6.03370361936173296419156773370, −4.57868719398359625773809237827, −3.46921390472154779558714393183, −2.05982911300005372440238464366, −0.991249617375358275808847787017, 1.56737211199009712052711009585, 2.78535449523025126146907050569, 3.62945880388350165985030654058, 4.93470656005361352619549228098, 5.71897997852795988648989929724, 7.40157058284550690891840097099, 7.87041739147204688865556471218, 8.761950501217224579229447681601, 9.257907767743186422549919248695, 10.31170054732168507667969524215

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