Properties

Label 2-882-63.5-c1-0-26
Degree $2$
Conductor $882$
Sign $0.992 - 0.125i$
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 + (1.54 − 0.780i)3-s − 4-s + (−0.183 − 0.317i)5-s + (0.780 + 1.54i)6-s i·8-s + (1.78 − 2.41i)9-s + (0.317 − 0.183i)10-s + (−0.579 − 0.334i)11-s + (−1.54 + 0.780i)12-s + (−0.867 − 0.500i)13-s + (−0.531 − 0.347i)15-s + 16-s + (2.49 + 4.32i)17-s + (2.41 + 1.78i)18-s + (5.50 + 3.17i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.892 − 0.450i)3-s − 0.5·4-s + (−0.0819 − 0.141i)5-s + (0.318 + 0.631i)6-s − 0.353i·8-s + (0.593 − 0.804i)9-s + (0.100 − 0.0579i)10-s + (−0.174 − 0.100i)11-s + (−0.446 + 0.225i)12-s + (−0.240 − 0.138i)13-s + (−0.137 − 0.0897i)15-s + 0.250·16-s + (0.605 + 1.04i)17-s + (0.569 + 0.419i)18-s + (1.26 + 0.729i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.992 - 0.125i$
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: \((2,\ 882,\ (\ :1/2),\ 0.992 - 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07192 + 0.130808i\)
\(L(\frac12)\) \(\approx\) \(2.07192 + 0.130808i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.54 + 0.780i)T \)
7 \( 1 \)
good5 \( 1 + (0.183 + 0.317i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.579 + 0.334i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.867 + 0.500i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.49 - 4.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.50 - 3.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.66 + 3.84i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.58 + 0.914i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
37 \( 1 + (-2.58 + 4.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.15 - 3.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.24 - 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.32T + 47T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.72T + 59T^{2} \)
61 \( 1 + 4.95iT - 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 - 5.49iT - 71T^{2} \)
73 \( 1 + (-3.52 + 2.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.35T + 79T^{2} \)
83 \( 1 + (-8.50 - 14.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.35 - 9.27i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.9 - 8.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.818297503754599462715682216044, −9.201360882125663871480982122809, −8.085668536775043996439532479539, −7.914914869406396362731832094344, −6.81151732371455233951099985092, −6.01107386466383809948871414623, −4.86607809604842474141058872614, −3.76641811351278576992725985801, −2.73796071969938767255466687906, −1.12076942665013579077635812358, 1.38361507077374396532515417267, 2.97437293112942028118089563887, 3.24182317015346594566907462402, 4.76538201052205210245061100614, 5.22908961300851847060651397009, 7.08168626311138839897826826706, 7.58205694754853586528827968746, 8.802767309971969291123063213411, 9.326168850278394029824411016817, 10.00742315735326526191762940362

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