Properties

Label 2-882-441.130-c1-0-46
Degree $2$
Conductor $882$
Sign $-0.721 + 0.692i$
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  + (−0.826 + 0.563i)2-s + (1.73 − 0.0218i)3-s + (0.365 − 0.930i)4-s + (−0.399 − 1.75i)5-s + (−1.41 + 0.993i)6-s + (−2.08 + 1.62i)7-s + (0.222 + 0.974i)8-s + (2.99 − 0.0755i)9-s + (1.31 + 1.22i)10-s + (−4.36 − 2.10i)11-s + (0.612 − 1.62i)12-s + (−5.61 + 3.82i)13-s + (0.808 − 2.51i)14-s + (−0.730 − 3.02i)15-s + (−0.733 − 0.680i)16-s + (−2.26 − 5.77i)17-s + ⋯
L(s)  = 1  + (−0.584 + 0.398i)2-s + (0.999 − 0.0125i)3-s + (0.182 − 0.465i)4-s + (−0.178 − 0.783i)5-s + (−0.579 + 0.405i)6-s + (−0.788 + 0.614i)7-s + (0.0786 + 0.344i)8-s + (0.999 − 0.0251i)9-s + (0.416 + 0.386i)10-s + (−1.31 − 0.633i)11-s + (0.176 − 0.467i)12-s + (−1.55 + 1.06i)13-s + (0.216 − 0.673i)14-s + (−0.188 − 0.781i)15-s + (−0.183 − 0.170i)16-s + (−0.550 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.721 + 0.692i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.721 + 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.163496 - 0.406457i\)
\(L(\frac12)\) \(\approx\) \(0.163496 - 0.406457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.826 - 0.563i)T \)
3 \( 1 + (-1.73 + 0.0218i)T \)
7 \( 1 + (2.08 - 1.62i)T \)
good5 \( 1 + (0.399 + 1.75i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (4.36 + 2.10i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (5.61 - 3.82i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (2.26 + 5.77i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (3.35 + 5.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.92 - 3.66i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-3.39 - 0.511i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (1.14 + 1.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (11.1 + 1.67i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (1.47 + 0.455i)T + (33.8 + 23.0i)T^{2} \)
43 \( 1 + (9.76 - 3.01i)T + (35.5 - 24.2i)T^{2} \)
47 \( 1 + (1.96 - 1.33i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-10.0 + 1.52i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-0.0711 + 0.0219i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-0.870 - 2.21i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-2.79 - 4.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.865 + 1.08i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (0.690 - 9.21i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-3.43 + 5.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.1 + 8.29i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (-4.38 - 2.99i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (-1.66 - 2.88i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.518865266391621571061941391428, −8.869011490449843194102312775176, −8.464083745250931201108910027328, −7.17383885570956505803917558335, −6.89520260534738341786277640141, −5.21600685474511570654597859077, −4.69358754473127632120993824949, −2.96231907949480280094393957908, −2.25772663742386960489604183965, −0.20135358769888503024266606309, 2.06135364312557487936829194463, 2.92923348762518166211831535203, 3.70355831647707230598993677041, 4.95105266025634263625083052079, 6.63798525130719073805111472054, 7.23850843419582515149634144613, 8.013091643281680884539535467921, 8.661138552878423161355509279846, 10.03166662110521289779906257782, 10.33163677127167022014783981373

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