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Properties

Label 2-847-11.4-c1-0-40
Degree $2$
Conductor $847$
Sign $0.975 - 0.220i$
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  + (1.03 + 0.750i)2-s + (−0.796 + 2.44i)3-s + (−0.114 − 0.352i)4-s + (3.31 − 2.40i)5-s + (−2.66 + 1.93i)6-s + (−0.309 − 0.951i)7-s + (0.935 − 2.87i)8-s + (−2.94 − 2.13i)9-s + 5.22·10-s + 0.954·12-s + (−3.55 − 2.58i)13-s + (0.394 − 1.21i)14-s + (3.25 + 10.0i)15-s + (2.52 − 1.83i)16-s + (3.39 − 2.46i)17-s + (−1.43 − 4.41i)18-s + ⋯
L(s)  = 1  + (0.730 + 0.530i)2-s + (−0.459 + 1.41i)3-s + (−0.0572 − 0.176i)4-s + (1.48 − 1.07i)5-s + (−1.08 + 0.789i)6-s + (−0.116 − 0.359i)7-s + (0.330 − 1.01i)8-s + (−0.980 − 0.712i)9-s + 1.65·10-s + 0.275·12-s + (−0.985 − 0.715i)13-s + (0.105 − 0.324i)14-s + (0.840 + 2.58i)15-s + (0.631 − 0.458i)16-s + (0.822 − 0.597i)17-s + (−0.338 − 1.04i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.975 - 0.220i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32663 + 0.259923i\)
\(L(\frac12)\) \(\approx\) \(2.32663 + 0.259923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-1.03 - 0.750i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.796 - 2.44i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-3.31 + 2.40i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (3.55 + 2.58i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.39 + 2.46i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.385 + 1.18i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.97T + 23T^{2} \)
29 \( 1 + (-0.598 - 1.84i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.26 - 0.918i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.221 + 0.681i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.48 - 4.57i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.35T + 43T^{2} \)
47 \( 1 + (3.23 - 9.95i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.21 - 2.33i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.25 - 13.0i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (9.48 - 6.89i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.59T + 67T^{2} \)
71 \( 1 + (0.176 - 0.128i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.39 + 10.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.69 + 2.68i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.98 + 1.44i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.20T + 89T^{2} \)
97 \( 1 + (-8.83 - 6.42i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.15869929769883091422384628081, −9.518143642761482735405622996388, −9.015717515317858270227055356213, −7.45760087866876681418507420495, −6.24640120028684169976141839452, −5.46932354450067096403911992328, −4.98416246848302818643058866836, −4.48669579498114093990475849441, −3.03551117597672862205478429176, −1.03032831869632850416012553339, 1.78342684171127579302616942600, 2.35147057200541274184708134193, 3.37134062583347348447905016288, 5.09601766768678894071629326448, 5.79211263146644808841295096292, 6.66626609371572489689668073215, 7.24752664826815024734189532105, 8.303440177473563845117348541763, 9.527503082270737348546471860057, 10.32794353833271203500154259650

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