Properties

Label 2-882-9.4-c1-0-25
Degree $2$
Conductor $882$
Sign $0.594 + 0.804i$
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 + (−1.18 − 1.26i)3-s + (−0.499 + 0.866i)4-s + (−0.686 + 1.18i)5-s + (0.5 − 1.65i)6-s − 0.999·8-s + (−0.186 + 2.99i)9-s − 1.37·10-s + (−2.18 − 3.78i)11-s + (1.68 − 0.396i)12-s + (1 − 1.73i)13-s + (2.31 − 0.543i)15-s + (−0.5 − 0.866i)16-s + 4.37·17-s + (−2.68 + 1.33i)18-s − 5·19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.684 − 0.728i)3-s + (−0.249 + 0.433i)4-s + (−0.306 + 0.531i)5-s + (0.204 − 0.677i)6-s − 0.353·8-s + (−0.0620 + 0.998i)9-s − 0.433·10-s + (−0.659 − 1.14i)11-s + (0.486 − 0.114i)12-s + (0.277 − 0.480i)13-s + (0.597 − 0.140i)15-s + (−0.125 − 0.216i)16-s + 1.06·17-s + (−0.633 + 0.314i)18-s − 1.14·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.594 + 0.804i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.594 + 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.905018 - 0.456734i\)
\(L(\frac12)\) \(\approx\) \(0.905018 - 0.456734i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 + 1.26i)T \)
7 \( 1 \)
good5 \( 1 + (0.686 - 1.18i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.18 + 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.37T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-3.68 + 6.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.37 + 2.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-5.18 + 8.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.55 + 7.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.74T + 53T^{2} \)
59 \( 1 + (-3.55 + 6.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.05 + 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.55 - 13.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.74 + 4.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.25T + 89T^{2} \)
97 \( 1 + (-4.55 - 7.89i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.39464900582480740732662937104, −8.780066708954714976301906405130, −8.083817375210178970275345099377, −7.36624592029444471228049046457, −6.48032582706566435465766279699, −5.79303530703679504942693763688, −5.00612751876087593455304802864, −3.64162141721823896375200343178, −2.54110089053881379362304815798, −0.51851934664225093502716716900, 1.32693635089189741031910185607, 2.97829142051715479256209401247, 4.17162404354524650813944691647, 4.75819663681857561496220244630, 5.57744561892566941631190842050, 6.61813138476618736180225682753, 7.77132788625453368588953569702, 8.889824971172755546992227980777, 9.675620035054704321902157602579, 10.29850023019607044120775905202

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