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Properties

Label 2-882-9.7-c1-0-1
Degree $2$
Conductor $882$
Sign $0.642 - 0.766i$
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  + (0.5 − 0.866i)2-s + (−1.22 − 1.22i)3-s + (−0.499 − 0.866i)4-s + (−0.965 − 1.67i)5-s + (−1.67 + 0.448i)6-s − 0.999·8-s + 2.99i·9-s − 1.93·10-s + (−2.73 + 4.73i)11-s + (−0.448 + 1.67i)12-s + (1.22 + 2.12i)13-s + (−0.866 + 3.23i)15-s + (−0.5 + 0.866i)16-s − 6.31·17-s + (2.59 + 1.49i)18-s + 1.93·19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.707 − 0.707i)3-s + (−0.249 − 0.433i)4-s + (−0.431 − 0.748i)5-s + (−0.683 + 0.183i)6-s − 0.353·8-s + 0.999i·9-s − 0.610·10-s + (−0.823 + 1.42i)11-s + (−0.129 + 0.482i)12-s + (0.339 + 0.588i)13-s + (−0.223 + 0.834i)15-s + (−0.125 + 0.216i)16-s − 1.53·17-s + (0.612 + 0.353i)18-s + 0.443·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.368556 + 0.171860i\)
\(L(\frac12)\) \(\approx\) \(0.368556 + 0.171860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
7 \( 1 \)
good5 \( 1 + (0.965 + 1.67i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.73 - 4.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.31T + 17T^{2} \)
19 \( 1 - 1.93T + 19T^{2} \)
23 \( 1 + (-2.96 - 5.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.366 + 0.633i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.67 + 6.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.901 + 1.56i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.76 - 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.26T + 53T^{2} \)
59 \( 1 + (-2.50 - 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.48 - 2.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.09 - 12.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 + 9.52T + 73T^{2} \)
79 \( 1 + (-5.06 + 8.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.94 - 8.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (-1.93 + 3.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.47329435446910039376978089095, −9.512872978046614713085531054983, −8.626004033255556198365828912948, −7.55144184941374092993743422930, −6.87779936737615474014285646175, −5.70254148730824790747402446593, −4.81076299250511453979991784970, −4.23324130601689966266853233039, −2.47479731859862725085543037822, −1.45715382172689641556711166794, 0.19229008274222679451276690285, 2.99019648552615722827457496569, 3.65755268697697360444525640320, 4.89135866590931845354295105562, 5.58077088786383350170820522006, 6.53252703119874800889964022714, 7.16448757463306832496692302247, 8.459269735230227333140765638973, 8.917182848576279213791252919739, 10.34923367160130076297414898938

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