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Properties

Label 2-882-63.5-c1-0-22
Degree $2$
Conductor $882$
Sign $0.240 - 0.970i$
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  + i·2-s + (1.52 − 0.819i)3-s − 4-s + (1.99 + 3.45i)5-s + (0.819 + 1.52i)6-s − i·8-s + (1.65 − 2.50i)9-s + (−3.45 + 1.99i)10-s + (1.43 + 0.828i)11-s + (−1.52 + 0.819i)12-s + (2.60 + 1.50i)13-s + (5.87 + 3.63i)15-s + 16-s + (−3.72 − 6.45i)17-s + (2.50 + 1.65i)18-s + (3.96 + 2.28i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.881 − 0.472i)3-s − 0.5·4-s + (0.891 + 1.54i)5-s + (0.334 + 0.623i)6-s − 0.353i·8-s + (0.552 − 0.833i)9-s + (−1.09 + 0.630i)10-s + (0.432 + 0.249i)11-s + (−0.440 + 0.236i)12-s + (0.723 + 0.417i)13-s + (1.51 + 0.938i)15-s + 0.250·16-s + (−0.904 − 1.56i)17-s + (0.589 + 0.390i)18-s + (0.908 + 0.524i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.240 - 0.970i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.240 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90961 + 1.49452i\)
\(L(\frac12)\) \(\approx\) \(1.90961 + 1.49452i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.52 + 0.819i)T \)
7 \( 1 \)
good5 \( 1 + (-1.99 - 3.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.43 - 0.828i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.60 - 1.50i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.72 + 6.45i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.96 - 2.28i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.253 + 0.146i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.18 - 1.83i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.71iT - 31T^{2} \)
37 \( 1 + (3.00 - 5.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.88 + 8.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.29 + 2.25i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 13.6T + 47T^{2} \)
53 \( 1 + (0.793 - 0.458i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.06T + 59T^{2} \)
61 \( 1 + 6.64iT - 61T^{2} \)
67 \( 1 - 4.05T + 67T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (-3.84 + 2.22i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.84T + 79T^{2} \)
83 \( 1 + (2.59 + 4.49i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.50 + 9.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.51 + 1.45i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.956717906191336434315069812701, −9.433241642489317289134439800115, −8.648024210542193969959004138624, −7.47110409091296521194048785998, −6.85967794961850438528637236972, −6.45242765385044566894261581153, −5.25506418012377193719572232639, −3.74254966552723564342583896003, −2.89326079596229700530030338549, −1.72942019129901903501551737285, 1.24544809777673193491415298150, 2.17211544281550388095551409765, 3.58828273931981767380222067660, 4.40566040077748372872052951427, 5.30056622763815120604273064830, 6.24430233640623456528879033578, 7.996327256660922044103224671409, 8.514220106905035133566355831747, 9.290266947490673391736272753549, 9.661800478981287764549675054429

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