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Properties

Label 2-882-147.104-c1-0-11
Degree $2$
Conductor $882$
Sign $0.349 + 0.936i$
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  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−1.69 − 0.817i)5-s + (0.171 − 2.64i)7-s + (−0.433 + 0.900i)8-s + (−0.817 − 1.69i)10-s + (2.06 + 1.64i)11-s + (−5.22 − 4.16i)13-s + (1.78 − 1.95i)14-s + (−0.900 + 0.433i)16-s + (0.735 − 3.22i)17-s − 2.04i·19-s + (0.419 − 1.83i)20-s + (0.587 + 2.57i)22-s + (3.23 − 0.737i)23-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (0.111 + 0.487i)4-s + (−0.759 − 0.365i)5-s + (0.0648 − 0.997i)7-s + (−0.153 + 0.318i)8-s + (−0.258 − 0.536i)10-s + (0.622 + 0.496i)11-s + (−1.44 − 1.15i)13-s + (0.475 − 0.523i)14-s + (−0.225 + 0.108i)16-s + (0.178 − 0.781i)17-s − 0.468i·19-s + (0.0937 − 0.410i)20-s + (0.125 + 0.549i)22-s + (0.673 − 0.153i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.349 + 0.936i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.349 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15590 - 0.802084i\)
\(L(\frac12)\) \(\approx\) \(1.15590 - 0.802084i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 - 0.623i)T \)
3 \( 1 \)
7 \( 1 + (-0.171 + 2.64i)T \)
good5 \( 1 + (1.69 + 0.817i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-2.06 - 1.64i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (5.22 + 4.16i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.735 + 3.22i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 2.04iT - 19T^{2} \)
23 \( 1 + (-3.23 + 0.737i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (1.74 + 0.399i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 4.79iT - 31T^{2} \)
37 \( 1 + (-0.420 + 1.84i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (0.136 + 0.0657i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-6.10 + 2.93i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (1.66 - 2.08i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-6.00 + 1.37i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-1.93 + 0.930i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-3.64 - 0.831i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + (7.79 - 1.77i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.54 + 2.03i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + (-7.36 - 9.23i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-5.52 - 6.92i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 12.7iT - 97T^{2} \)
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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

−9.949099755245726419789166016926, −9.112105021706867416794088182333, −7.83609896947354566288828211215, −7.50089635731880037137459944353, −6.72317065109785061694124220769, −5.36382411210672139067814886632, −4.60229405891134930717639105368, −3.85260315642461780339785323477, −2.63203438097081545779997948534, −0.55897490580775973543569210403, 1.76549919788513620105267044571, 2.94159146609778179735924339672, 3.90441110127506771139530465373, 4.86626279520079827323611822603, 5.86487032017495466447251898875, 6.78878134723783919595844391560, 7.66919320906936975346787175456, 8.805180973482083371375393232591, 9.434403518486222062657297182418, 10.46163612268215314678194980833

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