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Properties

Label 2-882-63.25-c1-0-38
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: $\chi_{882} (655, \cdot )$
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

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

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