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Properties

Label 2-882-7.2-c3-0-20
Degree $2$
Conductor $882$
Sign $0.991 + 0.126i$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (5.22 − 9.04i)5-s − 7.99·8-s + (−10.4 − 18.0i)10-s + (30.5 + 52.9i)11-s + 59.2·13-s + (−8 + 13.8i)16-s + (−10.2 − 17.7i)17-s + (−40.1 + 69.5i)19-s − 41.7·20-s + 122.·22-s + (−79.1 + 137. i)23-s + (7.93 + 13.7i)25-s + (59.2 − 102. i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.467 − 0.809i)5-s − 0.353·8-s + (−0.330 − 0.572i)10-s + (0.837 + 1.45i)11-s + 1.26·13-s + (−0.125 + 0.216i)16-s + (−0.145 − 0.252i)17-s + (−0.485 + 0.840i)19-s − 0.467·20-s + 1.18·22-s + (−0.717 + 1.24i)23-s + (0.0635 + 0.109i)25-s + (0.446 − 0.773i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.714611279\)
\(L(\frac12)\) \(\approx\) \(2.714611279\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-5.22 + 9.04i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-30.5 - 52.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 59.2T + 2.19e3T^{2} \)
17 \( 1 + (10.2 + 17.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (40.1 - 69.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (79.1 - 137. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 85.1T + 2.43e4T^{2} \)
31 \( 1 + (-121. - 210. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (145. - 251. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 168T + 6.89e4T^{2} \)
43 \( 1 - 7.62T + 7.95e4T^{2} \)
47 \( 1 + (-84.6 + 146. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (125. + 216. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-402. - 697. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-16.5 + 28.7i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-138. - 240. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 631.T + 3.57e5T^{2} \)
73 \( 1 + (384. + 665. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-209. + 362. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 761.T + 5.71e5T^{2} \)
89 \( 1 + (-786. + 1.36e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.04e3T + 9.12e5T^{2} \)
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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

−9.875477906891331131278374800872, −8.994916411981104548422029956119, −8.350226903288069323096836288338, −7.00886216713386781522717237065, −6.12480816069798079759084467502, −5.15446056322229204974905305603, −4.32242155744782223640721485438, −3.41800086905325087755954998731, −1.80823006419047680884452663954, −1.28084542155070143601927160119, 0.67242703785661933829848066769, 2.39431543388357397033533190537, 3.48674480269112863648041496033, 4.31505682379000445892581748990, 5.74517500537795526410812287736, 6.35364245824370778759414409201, 6.78778476005865526169660235814, 8.286126782493569435181190340487, 8.603003847388390827607578594571, 9.686163767712261436718406777151

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