Properties

Label 2-882-63.25-c1-0-19
Degree $2$
Conductor $882$
Sign $0.882 + 0.470i$
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  − 2-s + (−0.5 + 1.65i)3-s + 4-s + (2.18 − 3.78i)5-s + (0.5 − 1.65i)6-s − 8-s + (−2.5 − 1.65i)9-s + (−2.18 + 3.78i)10-s + (0.686 + 1.18i)11-s + (−0.5 + 1.65i)12-s + (1 + 1.73i)13-s + (5.18 + 5.51i)15-s + 16-s + (0.686 − 1.18i)17-s + (2.5 + 1.65i)18-s + (2.5 + 4.33i)19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.288 + 0.957i)3-s + 0.5·4-s + (0.977 − 1.69i)5-s + (0.204 − 0.677i)6-s − 0.353·8-s + (−0.833 − 0.552i)9-s + (−0.691 + 1.19i)10-s + (0.206 + 0.358i)11-s + (−0.144 + 0.478i)12-s + (0.277 + 0.480i)13-s + (1.33 + 1.42i)15-s + 0.250·16-s + (0.166 − 0.288i)17-s + (0.589 + 0.390i)18-s + (0.573 + 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.882 + 0.470i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.882 + 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15455 - 0.288739i\)
\(L(\frac12)\) \(\approx\) \(1.15455 - 0.288739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.5 - 1.65i)T \)
7 \( 1 \)
good5 \( 1 + (-2.18 + 3.78i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.686 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.686 + 1.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.813 + 1.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.37 + 7.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.31 - 4.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.05 + 7.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-4.37 + 7.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 + 3.11T + 61T^{2} \)
67 \( 1 + 2.11T + 67T^{2} \)
71 \( 1 + 7.11T + 71T^{2} \)
73 \( 1 + (-6.05 + 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.11T + 79T^{2} \)
83 \( 1 + (-8.74 + 15.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.37 - 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.05 - 7.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.873988632945888635580616574128, −9.301700989660957663051141532389, −8.739569911171878345078586387846, −7.898996862696681860123104950770, −6.38179615249510903771827153538, −5.64755148465717659849735431684, −4.84284062495842582755710351319, −3.90315696017910636251021454334, −2.16626085689598899485953502931, −0.851760292146256159865905463597, 1.29597139505715774102254452748, 2.55097256537709003880837194698, 3.21048608694487194339911942489, 5.46235514610557300153449040537, 6.11711128522442438731097350319, 6.96005265771186889141204050122, 7.34189556058844969738082064704, 8.487088882370984080837326764983, 9.412071610862862535758113609867, 10.36603229828982934804412217316

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