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Properties

Label 2-882-7.4-c3-0-29
Degree $2$
Conductor $882$
Sign $0.749 + 0.661i$
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 + (3.53 + 6.12i)5-s + 7.99·8-s + (7.07 − 12.2i)10-s + (20 − 34.6i)11-s + 63.6·13-s + (−8 − 13.8i)16-s + (−0.707 + 1.22i)17-s + (5.65 + 9.79i)19-s − 28.2·20-s − 80·22-s + (34 + 58.8i)23-s + (37.5 − 64.9i)25-s + (−63.6 − 110. i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.316 + 0.547i)5-s + 0.353·8-s + (0.223 − 0.387i)10-s + (0.548 − 0.949i)11-s + 1.35·13-s + (−0.125 − 0.216i)16-s + (−0.0100 + 0.0174i)17-s + (0.0683 + 0.118i)19-s − 0.316·20-s − 0.775·22-s + (0.308 + 0.533i)23-s + (0.299 − 0.519i)25-s + (−0.480 − 0.831i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.945599656\)
\(L(\frac12)\) \(\approx\) \(1.945599656\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.53 - 6.12i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-20 + 34.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 63.6T + 2.19e3T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-5.65 - 9.79i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-34 - 58.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 110T + 2.43e4T^{2} \)
31 \( 1 + (59.3 - 102. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-10 - 17.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 49.4T + 6.89e4T^{2} \)
43 \( 1 + 340T + 7.95e4T^{2} \)
47 \( 1 + (45.2 + 78.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-314 + 543. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-438. + 759. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-458. - 794. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (270 - 467. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 420T + 3.57e5T^{2} \)
73 \( 1 + (-144. + 251. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-380 - 658. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 944.T + 5.71e5T^{2} \)
89 \( 1 + (576. + 998. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 502.T + 9.12e5T^{2} \)
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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

−9.774014749954233522597335066325, −8.754509604626080787143116548530, −8.335095483377640158981269963876, −7.04063565500013781467642204196, −6.26979557701873366801873111207, −5.30112705625294879631106606089, −3.80784631563840860649979271410, −3.24842772335906478978031680372, −1.90665733730028605449084943467, −0.78554096857725900419915653388, 0.908654274835398097109291404080, 1.92830884183618383195272699126, 3.64697950930605026297558246514, 4.66166331138547206186628767969, 5.58278123075682828782450882202, 6.46324608941313022423697520897, 7.25151569135491398348337714893, 8.254980440894965040210465439814, 9.024831401709717808490578459525, 9.547842246050960439921398541633

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