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Properties

Label 2-882-441.101-c1-0-46
Degree $2$
Conductor $882$
Sign $-0.402 + 0.915i$
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 + (1.72 − 0.120i)3-s + (0.222 − 0.974i)4-s + (0.228 − 3.04i)5-s + (1.27 − 1.17i)6-s + (−2.63 − 0.175i)7-s + (−0.433 − 0.900i)8-s + (2.97 − 0.416i)9-s + (−1.72 − 2.52i)10-s + (0.0802 + 0.0314i)11-s + (0.267 − 1.71i)12-s + (−1.50 − 0.589i)13-s + (−2.17 + 1.50i)14-s + (0.0276 − 5.29i)15-s + (−0.900 − 0.433i)16-s + (2.58 + 0.796i)17-s + ⋯
L(s)  = 1  + (0.552 − 0.440i)2-s + (0.997 − 0.0695i)3-s + (0.111 − 0.487i)4-s + (0.102 − 1.36i)5-s + (0.520 − 0.478i)6-s + (−0.997 − 0.0663i)7-s + (−0.153 − 0.318i)8-s + (0.990 − 0.138i)9-s + (−0.544 − 0.799i)10-s + (0.0241 + 0.00949i)11-s + (0.0770 − 0.494i)12-s + (−0.416 − 0.163i)13-s + (−0.580 + 0.403i)14-s + (0.00713 − 1.36i)15-s + (−0.225 − 0.108i)16-s + (0.626 + 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.402 + 0.915i$
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: \((2,\ 882,\ (\ :1/2),\ -0.402 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46034 - 2.23636i\)
\(L(\frac12)\) \(\approx\) \(1.46034 - 2.23636i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.120i)T \)
7 \( 1 + (2.63 + 0.175i)T \)
good5 \( 1 + (-0.228 + 3.04i)T + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (-0.0802 - 0.0314i)T + (8.06 + 7.48i)T^{2} \)
13 \( 1 + (1.50 + 0.589i)T + (9.52 + 8.84i)T^{2} \)
17 \( 1 + (-2.58 - 0.796i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-0.583 + 0.336i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.293 - 0.316i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (-0.436 + 1.41i)T + (-23.9 - 16.3i)T^{2} \)
31 \( 1 + 7.23iT - 31T^{2} \)
37 \( 1 + (6.56 - 6.09i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (-1.70 - 1.16i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-4.69 + 3.20i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-1.27 - 1.60i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (6.96 - 7.50i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-6.60 - 3.18i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (7.13 - 1.62i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + (-8.90 - 2.03i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-4.84 + 1.90i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 - 4.41T + 79T^{2} \)
83 \( 1 + (-0.900 - 2.29i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (14.2 + 2.14i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + (-9.91 - 5.72i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.666983570929318994337846327482, −9.258113922144035630336082782856, −8.316938137112740961931279012399, −7.43520525691945046980600019248, −6.29180929894379916481406180935, −5.26013963126474649310394249926, −4.29814374881833054643842252935, −3.44194236157083576211708429695, −2.34353652690357305026605417498, −0.989344020897579201394348833747, 2.28394408036175455010893305506, 3.19652369503389493886741172463, 3.70670129059406957076338382808, 5.12672394724728590957133204511, 6.35597874508055790155418256678, 6.98737186328011685902604360612, 7.57330759726257154828307709998, 8.677470187134137514934215880197, 9.608994697005364759038236910721, 10.23841303892657102695271975091

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