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Properties

Label 2-882-21.20-c1-0-0
Degree $2$
Conductor $882$
Sign $-0.995 + 0.0980i$
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 − 4-s − 0.717·5-s − i·8-s − 0.717i·10-s + 3i·11-s − 2.44i·13-s + 16-s − 5.91·17-s + 5.91i·19-s + 0.717·20-s − 3·22-s + 4.24i·23-s − 4.48·25-s + 2.44·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.320·5-s − 0.353i·8-s − 0.226i·10-s + 0.904i·11-s − 0.679i·13-s + 0.250·16-s − 1.43·17-s + 1.35i·19-s + 0.160·20-s − 0.639·22-s + 0.884i·23-s − 0.897·25-s + 0.480·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.995 + 0.0980i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.995 + 0.0980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0280346 - 0.570252i\)
\(L(\frac12)\) \(\approx\) \(0.0280346 - 0.570252i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + 0.717T + 5T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + 5.91T + 17T^{2} \)
19 \( 1 - 5.91iT - 19T^{2} \)
23 \( 1 - 4.24iT - 23T^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + 9.08iT - 31T^{2} \)
37 \( 1 + 0.242T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 + 0.242T + 43T^{2} \)
47 \( 1 + 5.91T + 47T^{2} \)
53 \( 1 - 7.24iT - 53T^{2} \)
59 \( 1 - 8.06T + 59T^{2} \)
61 \( 1 + 1.01iT - 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + 1.43iT - 73T^{2} \)
79 \( 1 + 2.75T + 79T^{2} \)
83 \( 1 + 6.63T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−10.35592227203560011496717977173, −9.690895613793729834012088731491, −8.750456647455199800393788002497, −7.895662314751417095197825682938, −7.26636157553289107575935041670, −6.30662964856177018954704914913, −5.39525076564707989804185063717, −4.41685230965194117624707237538, −3.48995157079225962609680314174, −1.85586136600495704495656084185, 0.26326806086560037148294724131, 2.00552691845123296852125115724, 3.11767502456994874850778429874, 4.21556245666810672954805328143, 4.98268882295348862251349393880, 6.30876961478634493065263286809, 7.05472489745851859677858019540, 8.458813720151161865112782899079, 8.739168696634809321022170416047, 9.790522803334570655112082959601

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