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Properties

Label 2-847-11.3-c1-0-30
Degree $2$
Conductor $847$
Sign $0.642 + 0.766i$
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  + (1.10 − 0.801i)2-s + (0.112 + 0.345i)3-s + (−0.0436 + 0.134i)4-s + (−2.54 − 1.84i)5-s + (0.400 + 0.291i)6-s + (−0.309 + 0.951i)7-s + (0.902 + 2.77i)8-s + (2.32 − 1.68i)9-s − 4.28·10-s − 0.0513·12-s + (3.86 − 2.80i)13-s + (0.421 + 1.29i)14-s + (0.352 − 1.08i)15-s + (2.99 + 2.17i)16-s + (3.86 + 2.80i)17-s + (1.20 − 3.71i)18-s + ⋯
L(s)  = 1  + (0.779 − 0.566i)2-s + (0.0648 + 0.199i)3-s + (−0.0218 + 0.0672i)4-s + (−1.13 − 0.825i)5-s + (0.163 + 0.118i)6-s + (−0.116 + 0.359i)7-s + (0.318 + 0.981i)8-s + (0.773 − 0.561i)9-s − 1.35·10-s − 0.0148·12-s + (1.07 − 0.778i)13-s + (0.112 + 0.346i)14-s + (0.0910 − 0.280i)15-s + (0.747 + 0.543i)16-s + (0.937 + 0.681i)17-s + (0.284 − 0.876i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92341 - 0.897232i\)
\(L(\frac12)\) \(\approx\) \(1.92341 - 0.897232i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 + 0.801i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.112 - 0.345i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.54 + 1.84i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-3.86 + 2.80i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.86 - 2.80i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.16 + 6.66i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.14T + 23T^{2} \)
29 \( 1 + (-2.16 + 6.66i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.94 + 2.13i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.04 - 9.38i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.995 + 3.06i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.28T + 43T^{2} \)
47 \( 1 + (0.240 + 0.739i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.84 - 1.34i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.112 - 0.345i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.60 + 1.89i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.59T + 67T^{2} \)
71 \( 1 + (-12.2 - 8.91i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.995 - 3.06i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.00 + 2.18i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.25 + 0.914i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.58T + 89T^{2} \)
97 \( 1 + (4.97 - 3.61i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.28696259150721883618300635971, −9.067689298198018206527853359216, −8.392945858353108836292158747033, −7.76595671882171408290049007315, −6.49397560556433874625173534851, −5.25658104768219722739826949205, −4.43342979620918208096482226445, −3.76227871581430746418572998985, −2.88864972284101754209685018802, −1.02051229791062010534719239452, 1.33090415261922327390826413942, 3.33447439434881564170680588041, 3.97521620143277504918025810640, 4.88231333923083048780826054007, 6.05606726134003995619424047455, 6.97937700878263394669660365733, 7.35321422243948157598377932487, 8.296585174965363243489612022554, 9.552905417463463980116364001761, 10.58324318044664151848773053659

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