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Properties

Label 2-847-11.5-c1-0-16
Degree $2$
Conductor $847$
Sign $0.569 - 0.821i$
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  + (0.0699 + 0.215i)2-s + (0.177 + 0.128i)3-s + (1.57 − 1.14i)4-s + (−0.771 + 2.37i)5-s + (−0.0153 + 0.0471i)6-s + (−0.809 + 0.587i)7-s + (0.722 + 0.525i)8-s + (−0.912 − 2.80i)9-s − 0.564·10-s + 0.427·12-s + (1.58 + 4.88i)13-s + (−0.183 − 0.132i)14-s + (−0.442 + 0.321i)15-s + (1.14 − 3.51i)16-s + (−0.444 + 1.36i)17-s + (0.540 − 0.392i)18-s + ⋯
L(s)  = 1  + (0.0494 + 0.152i)2-s + (0.102 + 0.0743i)3-s + (0.788 − 0.572i)4-s + (−0.344 + 1.06i)5-s + (−0.00625 + 0.0192i)6-s + (−0.305 + 0.222i)7-s + (0.255 + 0.185i)8-s + (−0.304 − 0.935i)9-s − 0.178·10-s + 0.123·12-s + (0.440 + 1.35i)13-s + (−0.0489 − 0.0355i)14-s + (−0.114 + 0.0829i)15-s + (0.285 − 0.878i)16-s + (−0.107 + 0.331i)17-s + (0.127 − 0.0925i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.569 - 0.821i$
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: \((2,\ 847,\ (\ :1/2),\ 0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62499 + 0.850606i\)
\(L(\frac12)\) \(\approx\) \(1.62499 + 0.850606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0699 - 0.215i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.177 - 0.128i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.771 - 2.37i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-1.58 - 4.88i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.444 - 1.36i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.90 - 3.56i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 7.08T + 23T^{2} \)
29 \( 1 + (5.27 - 3.83i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.37 - 7.31i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.22 + 2.34i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.46 + 3.96i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 0.802T + 43T^{2} \)
47 \( 1 + (5.46 + 3.96i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.03 - 6.25i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.32 - 1.69i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.264 - 0.813i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.64T + 67T^{2} \)
71 \( 1 + (-1.39 + 4.29i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-12.0 + 8.72i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.757 - 2.33i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.694 + 2.13i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 1.73T + 89T^{2} \)
97 \( 1 + (3.73 + 11.4i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.44338261706355875698207315456, −9.490324967429461502496272536679, −8.787682160321270413756776409635, −7.39955032397432250588137645856, −6.80482541857555130180290971080, −6.24522769066793975583666859544, −5.19748964152107389311268686326, −3.62713607205946630395346168384, −2.98063687590258188950205496391, −1.52247405714516289531394239839, 0.947474673362749428629113298829, 2.57360155319375862905827496537, 3.42845277584715912996470831161, 4.73192064132566349306473439305, 5.51473453755179522099550106412, 6.75178840291175866427902015806, 7.80033884468193874002801258644, 8.077989669902893378480820017637, 9.135388053807010266862151235966, 10.11172101428382109140563894749

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