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Properties

Label 2-882-21.2-c2-0-25
Degree $2$
Conductor $882$
Sign $0.645 - 0.763i$
Analytic cond. $24.0327$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (7.70 + 4.44i)5-s + 2.82i·8-s + (6.29 + 10.8i)10-s + (15.4 − 8.89i)11-s − 2.58·13-s + (−2.00 + 3.46i)16-s + (22.4 − 12.9i)17-s + (10 − 17.3i)19-s + 17.7i·20-s + 25.1·22-s + (−15.4 − 8.89i)23-s + (27.0 + 46.9i)25-s + (−3.16 − 1.82i)26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (1.54 + 0.889i)5-s + 0.353i·8-s + (0.629 + 1.08i)10-s + (1.40 − 0.808i)11-s − 0.198·13-s + (−0.125 + 0.216i)16-s + (1.31 − 0.760i)17-s + (0.526 − 0.911i)19-s + 0.889i·20-s + 1.14·22-s + (−0.670 − 0.386i)23-s + (1.08 + 1.87i)25-s + (−0.121 − 0.0702i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.645 - 0.763i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ 0.645 - 0.763i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.000736247\)
\(L(\frac12)\) \(\approx\) \(4.000736247\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-7.70 - 4.44i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-15.4 + 8.89i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 2.58T + 169T^{2} \)
17 \( 1 + (-22.4 + 12.9i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-10 + 17.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (15.4 + 8.89i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 11.9iT - 841T^{2} \)
31 \( 1 + (8.58 + 14.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (19 - 32.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 15.7iT - 1.68e3T^{2} \)
43 \( 1 + 43.4T + 1.84e3T^{2} \)
47 \( 1 + (14.6 + 8.48i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (74.0 - 42.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (1.42 - 0.824i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-50.1 + 86.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (18.3 + 31.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 17.7iT - 5.04e3T^{2} \)
73 \( 1 + (-14.4 - 25.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (59.1 - 102. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 120. iT - 6.88e3T^{2} \)
89 \( 1 + (120. + 69.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 44.4T + 9.40e3T^{2} \)
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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

−9.832233565260889144620998816960, −9.483625481643919125259319868101, −8.337217406223802279344043807972, −7.09034740222175667459178008365, −6.52442898547670128642691042022, −5.79271449543321063388704996241, −4.99189750295364214263814874445, −3.50499453624979192962025947891, −2.75953883772859094022213133565, −1.41718034698154558609211483954, 1.39704930623650711958903272289, 1.84757359834121864679623586142, 3.47484556227902221075493953512, 4.48564615450240993311297841411, 5.53433368392839070356050668511, 5.97728419645056686259682409838, 7.02015501448507554699386546239, 8.280886343146480219676004016644, 9.343319353943428587697806232150, 9.820790857701581638399678207061

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