Properties

Label 2-847-77.4-c1-0-59
Degree $2$
Conductor $847$
Sign $-0.321 - 0.946i$
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  + (0.191 − 1.82i)2-s + (−1.47 − 1.63i)3-s + (−1.33 − 0.283i)4-s + (−0.580 − 0.258i)5-s + (−3.26 + 2.37i)6-s + (2.39 − 1.11i)7-s + (0.360 − 1.10i)8-s + (−0.191 + 1.82i)9-s + (−0.582 + 1.00i)10-s + (1.5 + 2.59i)12-s + (−1.45 − 1.05i)13-s + (−1.57 − 4.59i)14-s + (0.431 + 1.32i)15-s + (−4.44 − 1.97i)16-s + (−0.296 − 2.81i)17-s + (3.29 + 0.699i)18-s + ⋯
L(s)  = 1  + (0.135 − 1.28i)2-s + (−0.849 − 0.943i)3-s + (−0.667 − 0.141i)4-s + (−0.259 − 0.115i)5-s + (−1.33 + 0.967i)6-s + (0.907 − 0.421i)7-s + (0.127 − 0.391i)8-s + (−0.0639 + 0.608i)9-s + (−0.184 + 0.319i)10-s + (0.433 + 0.749i)12-s + (−0.404 − 0.293i)13-s + (−0.420 − 1.22i)14-s + (0.111 + 0.343i)15-s + (−1.11 − 0.494i)16-s + (−0.0718 − 0.683i)17-s + (0.775 + 0.164i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.321 - 0.946i$
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: \((2,\ 847,\ (\ :1/2),\ -0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.567911 + 0.792487i\)
\(L(\frac12)\) \(\approx\) \(0.567911 + 0.792487i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.39 + 1.11i)T \)
11 \( 1 \)
good2 \( 1 + (-0.191 + 1.82i)T + (-1.95 - 0.415i)T^{2} \)
3 \( 1 + (1.47 + 1.63i)T + (-0.313 + 2.98i)T^{2} \)
5 \( 1 + (0.580 + 0.258i)T + (3.34 + 3.71i)T^{2} \)
13 \( 1 + (1.45 + 1.05i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.296 + 2.81i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (5.44 - 1.15i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (1.08 + 1.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.22 - 9.91i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.87 + 2.61i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-4.05 + 4.50i)T + (-3.86 - 36.7i)T^{2} \)
41 \( 1 + (2.32 - 7.16i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.86T + 43T^{2} \)
47 \( 1 + (-2.77 + 0.589i)T + (42.9 - 19.1i)T^{2} \)
53 \( 1 + (-6.82 + 3.03i)T + (35.4 - 39.3i)T^{2} \)
59 \( 1 + (11.5 + 2.45i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (-3.95 - 1.76i)T + (40.8 + 45.3i)T^{2} \)
67 \( 1 + (-0.801 + 1.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.47 - 2.52i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (15.6 + 3.32i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (-0.497 + 4.73i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (7.46 - 5.42i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.182 + 0.315i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.10 - 1.52i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.08610293115486113146397593259, −8.822957689367216757137375760913, −7.79239958025166076475323761823, −7.03187752276798466215278717627, −6.13570565609211537173008153240, −4.86000820281074450070151309076, −4.12200054558546764479870030002, −2.65664175200191083203277667514, −1.57594831552389736102466087283, −0.50793947432956790105375344217, 2.21240589490226800779003607244, 4.28663352352746340415933292173, 4.60348599439188198001003139194, 5.72058125587844047580369639788, 6.14277375949301595936347171538, 7.32643851677349390559071962292, 8.119854986087729526891912560485, 8.838400223601673511897570665224, 10.00286836694432510214587553618, 10.78573494515355923555615280034

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