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Properties

Label 2-882-441.101-c1-0-45
Degree $2$
Conductor $882$
Sign $-0.923 + 0.383i$
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 + (1.51 − 0.843i)3-s + (0.222 − 0.974i)4-s + (−0.0410 + 0.548i)5-s + (−0.656 + 1.60i)6-s + (−2.05 + 1.67i)7-s + (0.433 + 0.900i)8-s + (1.57 − 2.55i)9-s + (−0.309 − 0.454i)10-s + (−2.37 − 0.931i)11-s + (−0.485 − 1.66i)12-s + (−5.06 − 1.98i)13-s + (0.561 − 2.58i)14-s + (0.400 + 0.864i)15-s + (−0.900 − 0.433i)16-s + (−6.80 − 2.09i)17-s + ⋯
L(s)  = 1  + (−0.552 + 0.440i)2-s + (0.873 − 0.487i)3-s + (0.111 − 0.487i)4-s + (−0.0183 + 0.245i)5-s + (−0.268 + 0.654i)6-s + (−0.775 + 0.631i)7-s + (0.153 + 0.318i)8-s + (0.525 − 0.850i)9-s + (−0.0979 − 0.143i)10-s + (−0.715 − 0.280i)11-s + (−0.140 − 0.479i)12-s + (−1.40 − 0.551i)13-s + (0.150 − 0.691i)14-s + (0.103 + 0.223i)15-s + (−0.225 − 0.108i)16-s + (−1.65 − 0.509i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.923 + 0.383i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0319904 - 0.160317i\)
\(L(\frac12)\) \(\approx\) \(0.0319904 - 0.160317i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.51 + 0.843i)T \)
7 \( 1 + (2.05 - 1.67i)T \)
good5 \( 1 + (0.0410 - 0.548i)T + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (2.37 + 0.931i)T + (8.06 + 7.48i)T^{2} \)
13 \( 1 + (5.06 + 1.98i)T + (9.52 + 8.84i)T^{2} \)
17 \( 1 + (6.80 + 2.09i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (4.29 - 2.47i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.35 - 3.61i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (1.39 - 4.52i)T + (-23.9 - 16.3i)T^{2} \)
31 \( 1 + 9.09iT - 31T^{2} \)
37 \( 1 + (-6.97 + 6.46i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (-0.309 - 0.210i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-0.963 + 0.657i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-7.82 - 9.80i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (2.37 - 2.55i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (7.53 + 3.62i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (7.49 - 1.71i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + 1.93T + 67T^{2} \)
71 \( 1 + (-3.73 - 0.851i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-3.70 + 1.45i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + 8.82T + 79T^{2} \)
83 \( 1 + (-1.93 - 4.93i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (11.1 + 1.68i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + (-3.90 - 2.25i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.400313426349533905173525846798, −9.064637789566567236192556761551, −7.953232179148261980454850477256, −7.42190497586868573236578155081, −6.50148583000590619523934212750, −5.69836158607057815419399078583, −4.32907054372407226141514749343, −2.81520922399261155791632931244, −2.24893687277774296893312670782, −0.07460348080742589978252459986, 2.13978158633864180236368506485, 2.84379170921256640916653157998, 4.27860805267804567426831222658, 4.66293896877499199632665853209, 6.57356221605656997715127815519, 7.23085242104920334747151933210, 8.255477088809939583733028172243, 8.927017537847739221914525873835, 9.666896433165943104424509835526, 10.40957494121351450561950082590

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