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Properties

Label 2-847-11.5-c1-0-27
Degree $2$
Conductor $847$
Sign $0.598 - 0.801i$
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  + (0.428 + 1.31i)2-s + (0.0990 + 0.0719i)3-s + (0.0640 − 0.0465i)4-s + (−0.0411 + 0.126i)5-s + (−0.0524 + 0.161i)6-s + (0.809 − 0.587i)7-s + (2.33 + 1.69i)8-s + (−0.922 − 2.83i)9-s − 0.184·10-s + 0.00969·12-s + (0.198 + 0.610i)13-s + (1.12 + 0.814i)14-s + (−0.0131 + 0.00957i)15-s + (−1.18 + 3.64i)16-s + (0.440 − 1.35i)17-s + (3.34 − 2.43i)18-s + ⋯
L(s)  = 1  + (0.302 + 0.932i)2-s + (0.0572 + 0.0415i)3-s + (0.0320 − 0.0232i)4-s + (−0.0183 + 0.0565i)5-s + (−0.0214 + 0.0659i)6-s + (0.305 − 0.222i)7-s + (0.824 + 0.598i)8-s + (−0.307 − 0.946i)9-s − 0.0582·10-s + 0.00279·12-s + (0.0549 + 0.169i)13-s + (0.299 + 0.217i)14-s + (−0.00340 + 0.00247i)15-s + (−0.296 + 0.911i)16-s + (0.106 − 0.328i)17-s + (0.788 − 0.573i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.598 - 0.801i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.598 - 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00910 + 1.00722i\)
\(L(\frac12)\) \(\approx\) \(2.00910 + 1.00722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-0.428 - 1.31i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.0990 - 0.0719i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.0411 - 0.126i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.198 - 0.610i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.440 + 1.35i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.81 - 4.22i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.66T + 23T^{2} \)
29 \( 1 + (-3.62 + 2.63i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.11 + 6.49i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.04 - 2.21i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.99 - 3.62i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.03T + 43T^{2} \)
47 \( 1 + (7.67 + 5.57i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.205 + 0.633i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.62 - 4.81i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.93 - 9.03i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + (-1.49 + 4.59i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.16 - 5.20i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.55 + 10.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.96 - 9.11i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 + (-2.03 - 6.25i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.21079243983593171230250286592, −9.423326214023621755399909107063, −8.401361914391908132554483884476, −7.57052805432055761407095183707, −6.86045248623327741514621861046, −5.97476090979254612420054056199, −5.24775313931620050332187065501, −4.16286533065202407569584354082, −2.97209398154825288107049195356, −1.30966062161842436515226596979, 1.32422210878196936309533017552, 2.56131737533521051016368857976, 3.30310730641469721011151235765, 4.67286777525040164949962070366, 5.28310430064447221817170163354, 6.72752985435579861086459443813, 7.54862423640982625624941808493, 8.399288568033429786778100879404, 9.354420006891383129211618959714, 10.42513897683303453364745608429

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