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Properties

Label 2-882-63.59-c1-0-15
Degree $2$
Conductor $882$
Sign $0.500 - 0.865i$
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  + (−0.866 − 0.5i)2-s + (1.44 + 0.948i)3-s + (0.499 + 0.866i)4-s − 0.366·5-s + (−0.780 − 1.54i)6-s − 0.999i·8-s + (1.20 + 2.74i)9-s + (0.317 + 0.183i)10-s + 0.669i·11-s + (−0.0967 + 1.72i)12-s + (0.867 + 0.500i)13-s + (−0.531 − 0.347i)15-s + (−0.5 + 0.866i)16-s + (−2.49 + 4.32i)17-s + (0.334 − 2.98i)18-s + (5.50 − 3.17i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.836 + 0.547i)3-s + (0.249 + 0.433i)4-s − 0.163·5-s + (−0.318 − 0.631i)6-s − 0.353i·8-s + (0.400 + 0.916i)9-s + (0.100 + 0.0579i)10-s + 0.201i·11-s + (−0.0279 + 0.499i)12-s + (0.240 + 0.138i)13-s + (−0.137 − 0.0897i)15-s + (−0.125 + 0.216i)16-s + (−0.605 + 1.04i)17-s + (0.0788 − 0.702i)18-s + (1.26 − 0.729i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.500 - 0.865i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.500 - 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26177 + 0.727940i\)
\(L(\frac12)\) \(\approx\) \(1.26177 + 0.727940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.44 - 0.948i)T \)
7 \( 1 \)
good5 \( 1 + 0.366T + 5T^{2} \)
11 \( 1 - 0.669iT - 11T^{2} \)
13 \( 1 + (-0.867 - 0.500i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.49 - 4.32i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.50 + 3.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.69iT - 23T^{2} \)
29 \( 1 + (-1.58 + 0.914i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.47 + 3.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.58 - 4.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.15 + 3.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.24 - 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.16 - 7.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.36 + 7.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.29 + 2.47i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.44 - 9.43i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.49iT - 71T^{2} \)
73 \( 1 + (-3.52 - 2.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.17 - 7.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.50 + 14.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.35 - 9.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.9 + 8.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.853034234368576785904186938859, −9.665917708465589978784684955008, −8.667999932883394543221887550530, −7.935821668188032810598245348435, −7.26198668269736992244302759808, −5.98311411746212059115629855773, −4.65843375647804424415094329619, −3.74536690302183263993304472236, −2.79270594883393887433676843492, −1.56302071255170007141131614629, 0.824948009860976761735078269028, 2.27453625008116255953639916551, 3.32852019332446253529027813140, 4.63481557559165536004459240930, 5.95288846270542857805898265308, 6.79598499999504711197540602506, 7.60233215379320556669637083384, 8.263257796402014798708647531834, 9.029511956290882839961055022897, 9.748895170358973216614539543379

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