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Properties

Label 2-882-7.5-c2-0-29
Degree $2$
Conductor $882$
Sign $-0.832 + 0.553i$
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  + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (4.24 + 2.44i)5-s − 2.82·8-s + (6 − 3.46i)10-s + (−8.48 − 14.6i)11-s − 1.73i·13-s + (−2.00 + 3.46i)16-s + (−4.24 + 2.44i)17-s + (−25.5 − 14.7i)19-s − 9.79i·20-s − 24·22-s + (−4.24 + 7.34i)23-s + (−0.500 − 0.866i)25-s + (−2.12 − 1.22i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.848 + 0.489i)5-s − 0.353·8-s + (0.600 − 0.346i)10-s + (−0.771 − 1.33i)11-s − 0.133i·13-s + (−0.125 + 0.216i)16-s + (−0.249 + 0.144i)17-s + (−1.34 − 0.774i)19-s − 0.489i·20-s − 1.09·22-s + (−0.184 + 0.319i)23-s + (−0.0200 − 0.0346i)25-s + (−0.0815 − 0.0471i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ -0.832 + 0.553i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.648322707\)
\(L(\frac12)\) \(\approx\) \(1.648322707\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-4.24 - 2.44i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.48 + 14.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 1.73iT - 169T^{2} \)
17 \( 1 + (4.24 - 2.44i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (25.5 + 14.7i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (4.24 - 7.34i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 33.9T + 841T^{2} \)
31 \( 1 + (-10.5 + 6.06i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-23.5 + 40.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 68.5iT - 1.68e3T^{2} \)
43 \( 1 - 31T + 1.84e3T^{2} \)
47 \( 1 + (72.1 + 41.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (38.1 + 66.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (72.1 - 41.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-72 - 41.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-15.5 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 59.3T + 5.04e3T^{2} \)
73 \( 1 + (-70.5 + 40.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (20.5 - 35.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 4.89iT - 6.88e3T^{2} \)
89 \( 1 + (-50.9 - 29.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 41.5iT - 9.40e3T^{2} \)
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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

−9.823501503405758859196112333164, −8.843324776131271272384020607023, −8.118157641713158071566639331076, −6.72290373123749639508824393835, −6.01543365312909813469146378450, −5.22570592076553462478790842951, −4.03516007467410770028037376493, −2.86332636835051522706833611320, −2.13940729823331137965930335936, −0.44667277464964564132988662638, 1.66866018634715994168469141536, 2.80227115973976438946350686851, 4.47801955714005137688225392233, 4.84410008729259958665389450979, 6.06783796523835425224050753694, 6.61449082392830991980332719153, 7.80986874173903557310189319571, 8.403789291905795922106708505188, 9.559805329814295713879461704583, 9.986635555328119966898536240289

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