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Properties

Label 2-847-77.4-c1-0-32
Degree $2$
Conductor $847$
Sign $-0.335 - 0.941i$
Analytic cond. $6.76332$
Root an. cond. $2.60064$
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  + (−0.0704 + 0.670i)2-s + (1.43 + 1.59i)3-s + (1.51 + 0.321i)4-s + (1.54 + 0.687i)5-s + (−1.17 + 0.852i)6-s + (1.55 + 2.14i)7-s + (−0.738 + 2.27i)8-s + (−0.170 + 1.61i)9-s + (−0.569 + 0.985i)10-s + (1.66 + 2.87i)12-s + (−3.21 − 2.33i)13-s + (−1.54 + 0.891i)14-s + (1.12 + 3.45i)15-s + (1.35 + 0.602i)16-s + (−0.268 − 2.55i)17-s + (−1.07 − 0.228i)18-s + ⋯
L(s)  = 1  + (−0.0498 + 0.473i)2-s + (0.831 + 0.922i)3-s + (0.755 + 0.160i)4-s + (0.690 + 0.307i)5-s + (−0.478 + 0.347i)6-s + (0.587 + 0.808i)7-s + (−0.261 + 0.803i)8-s + (−0.0567 + 0.539i)9-s + (−0.179 + 0.311i)10-s + (0.479 + 0.831i)12-s + (−0.891 − 0.647i)13-s + (−0.412 + 0.238i)14-s + (0.289 + 0.892i)15-s + (0.338 + 0.150i)16-s + (−0.0651 − 0.619i)17-s + (−0.252 − 0.0537i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.335 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.335 - 0.941i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.335 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58216 + 2.24371i\)
\(L(\frac12)\) \(\approx\) \(1.58216 + 2.24371i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.55 - 2.14i)T \)
11 \( 1 \)
good2 \( 1 + (0.0704 - 0.670i)T + (-1.95 - 0.415i)T^{2} \)
3 \( 1 + (-1.43 - 1.59i)T + (-0.313 + 2.98i)T^{2} \)
5 \( 1 + (-1.54 - 0.687i)T + (3.34 + 3.71i)T^{2} \)
13 \( 1 + (3.21 + 2.33i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.268 + 2.55i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (-0.326 + 0.0694i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (2.11 + 3.66i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.961 + 2.96i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.98 - 3.11i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-6.78 + 7.53i)T + (-3.86 - 36.7i)T^{2} \)
41 \( 1 + (-3.34 + 10.3i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.26T + 43T^{2} \)
47 \( 1 + (-2.31 + 0.491i)T + (42.9 - 19.1i)T^{2} \)
53 \( 1 + (4.00 - 1.78i)T + (35.4 - 39.3i)T^{2} \)
59 \( 1 + (-0.0342 - 0.00727i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (-3.75 - 1.67i)T + (40.8 + 45.3i)T^{2} \)
67 \( 1 + (2.72 - 4.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.62 - 5.54i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.293 + 0.0623i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (1.48 - 14.1i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (3.75 - 2.72i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (6.99 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.67 - 1.94i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.34098631066936120685542087580, −9.475183858910965532894179694143, −8.824444757106725427923414920643, −7.929678147012285257408237854108, −7.17269712058450191380611516321, −5.94532690993389086428454470388, −5.34840856801616259119323301621, −4.09483880738828349155295385303, −2.65726860876180767478427550913, −2.33476606878515841092303869475, 1.43848648319058512189549056188, 1.91520515505015408841363777525, 3.04320201138148886545589100846, 4.34713338872540104980236235980, 5.67945400300490110069881400171, 6.69824439646030026363483475892, 7.48716803580146142838255687263, 7.983060949478525963672516914334, 9.282014317940789327553637824198, 9.836156008518854040962166975068

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