Properties

Label 2-882-63.58-c1-0-12
Degree $2$
Conductor $882$
Sign $-0.574 - 0.818i$
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 + (−0.724 + 1.57i)3-s + 4-s + (1.72 + 2.98i)5-s + (−0.724 + 1.57i)6-s + 8-s + (−1.94 − 2.28i)9-s + (1.72 + 2.98i)10-s + (−1 + 1.73i)11-s + (−0.724 + 1.57i)12-s + (−2.44 + 4.24i)13-s + (−5.94 + 0.548i)15-s + 16-s + (1 + 1.73i)17-s + (−1.94 − 2.28i)18-s + (3.72 − 6.45i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.418 + 0.908i)3-s + 0.5·4-s + (0.771 + 1.33i)5-s + (−0.295 + 0.642i)6-s + 0.353·8-s + (−0.649 − 0.760i)9-s + (0.545 + 0.944i)10-s + (−0.301 + 0.522i)11-s + (−0.209 + 0.454i)12-s + (−0.679 + 1.17i)13-s + (−1.53 + 0.141i)15-s + 0.250·16-s + (0.242 + 0.420i)17-s + (−0.459 − 0.537i)18-s + (0.854 − 1.48i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.574 - 0.818i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.574 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995093 + 1.91372i\)
\(L(\frac12)\) \(\approx\) \(0.995093 + 1.91372i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.724 - 1.57i)T \)
7 \( 1 \)
good5 \( 1 + (-1.72 - 2.98i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.44 - 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.72 + 6.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.44 + 2.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + (-3.89 + 6.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.89 - 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.44 - 2.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + (-0.550 - 0.953i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 3.10T + 67T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 + (-1.44 - 2.51i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 7.89T + 79T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.55 - 6.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.44 + 5.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.57731922503000535545661082985, −9.659837607360981130934642208178, −9.252461974045078832473707520411, −7.47421055845399518770852833030, −6.81281827754723879696805369568, −5.99303144125706068855588775838, −5.13005932000762808024449690373, −4.22220973266364490290936856149, −3.09880997328276191471419829720, −2.21852798134589504987379615172, 0.864340788097194765073690537970, 2.01837700243424224814722347538, 3.30374751120496943846137109414, 4.92680244568265268925859438190, 5.51021321876497266921793463478, 5.94105885439756850034183886156, 7.31034161089436904541696375998, 7.964892383204558731544363354834, 8.879497509456085088645711759211, 9.975181730277029025981319197485

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