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Properties

Label 2-882-9.7-c1-0-27
Degree $2$
Conductor $882$
Sign $0.704 + 0.709i$
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 + (1.73 + 0.0789i)3-s + (−0.499 − 0.866i)4-s + (0.296 + 0.514i)5-s + (0.933 − 1.45i)6-s − 0.999·8-s + (2.98 + 0.273i)9-s + 0.593·10-s + (0.296 − 0.514i)11-s + (−0.796 − 1.53i)12-s + (1.25 + 2.17i)13-s + (0.472 + 0.912i)15-s + (−0.5 + 0.866i)16-s + 2.92·17-s + (1.73 − 2.45i)18-s + 5.38·19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.998 + 0.0455i)3-s + (−0.249 − 0.433i)4-s + (0.132 + 0.229i)5-s + (0.381 − 0.595i)6-s − 0.353·8-s + (0.995 + 0.0910i)9-s + 0.187·10-s + (0.0894 − 0.154i)11-s + (−0.230 − 0.443i)12-s + (0.348 + 0.603i)13-s + (0.122 + 0.235i)15-s + (−0.125 + 0.216i)16-s + 0.708·17-s + (0.407 − 0.577i)18-s + 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.704 + 0.709i$
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: \((2,\ 882,\ (\ :1/2),\ 0.704 + 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56308 - 1.06749i\)
\(L(\frac12)\) \(\approx\) \(2.56308 - 1.06749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 0.0789i)T \)
7 \( 1 \)
good5 \( 1 + (-0.296 - 0.514i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.296 + 0.514i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.25 - 2.17i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.92T + 17T^{2} \)
19 \( 1 - 5.38T + 19T^{2} \)
23 \( 1 + (2.23 + 3.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.09 - 5.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.93 + 6.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-0.136 - 0.236i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.58 - 9.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.08 + 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.05T + 53T^{2} \)
59 \( 1 + (-4.32 - 7.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.32 - 5.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.956 - 1.65i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 - 7.91T + 73T^{2} \)
79 \( 1 + (-4.62 + 8.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.85 - 6.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.984045403655315261921234758276, −9.318578679907770784473199229931, −8.516318343312860053641882993312, −7.57556499777794848675934168032, −6.63273692560617424120641585000, −5.50203893443815254269545757460, −4.35135421270764634250426321897, −3.48983487676545225177100808377, −2.59102841961004127645758767435, −1.39729935894885690216774863913, 1.50241951161041980747433022342, 3.09403179424567646430434645773, 3.75091300814333558663177931802, 5.02857128555763209823286441274, 5.79724517575406602682864628723, 7.08016025418009295341250204763, 7.65357112062730812097652964305, 8.423729679627611624056151246606, 9.334689732826509625499229926444, 9.869210347615748941292725056198

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