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Properties

Label 2-847-11.4-c1-0-11
Degree $2$
Conductor $847$
Sign $0.530 + 0.847i$
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  + (−1.38 − 1.00i)2-s + (0.708 − 2.17i)3-s + (0.282 + 0.869i)4-s + (−3.28 + 2.39i)5-s + (−3.16 + 2.29i)6-s + (−0.309 − 0.951i)7-s + (−0.572 + 1.76i)8-s + (−1.82 − 1.32i)9-s + 6.94·10-s + 2.09·12-s + (2.65 + 1.92i)13-s + (−0.527 + 1.62i)14-s + (2.87 + 8.86i)15-s + (4.03 − 2.93i)16-s + (−1.06 + 0.776i)17-s + (1.18 + 3.65i)18-s + ⋯
L(s)  = 1  + (−0.976 − 0.709i)2-s + (0.408 − 1.25i)3-s + (0.141 + 0.434i)4-s + (−1.47 + 1.06i)5-s + (−1.29 + 0.938i)6-s + (−0.116 − 0.359i)7-s + (−0.202 + 0.623i)8-s + (−0.607 − 0.441i)9-s + 2.19·10-s + 0.604·12-s + (0.735 + 0.534i)13-s + (−0.140 + 0.433i)14-s + (0.743 + 2.28i)15-s + (1.00 − 0.733i)16-s + (−0.259 + 0.188i)17-s + (0.279 + 0.861i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.530 + 0.847i$
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: \((2,\ 847,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.592851 - 0.328514i\)
\(L(\frac12)\) \(\approx\) \(0.592851 - 0.328514i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 + 1.00i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.708 + 2.17i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (3.28 - 2.39i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-2.65 - 1.92i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.06 - 0.776i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.668 - 2.05i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.86T + 23T^{2} \)
29 \( 1 + (-0.0754 - 0.232i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.55 - 4.03i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.0789 - 0.243i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.77 + 5.45i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.01T + 43T^{2} \)
47 \( 1 + (1.25 - 3.86i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.04 - 2.94i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.303 + 0.935i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.49 + 1.08i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 3.00T + 67T^{2} \)
71 \( 1 + (5.23 - 3.80i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.98 - 9.17i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.47 + 3.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.67 + 1.21i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + (-2.09 - 1.51i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.36624870816496057849811607802, −9.018674374164582130106471964407, −8.242775802620199279801111904308, −7.71715860733799687748254761349, −6.96849345329088890586758150205, −6.16595904302943122056594324651, −4.23058088533322786746361132721, −3.18521040383546517878942391623, −2.19199840765093534548199010235, −0.894176836684953441642226649140, 0.63945645700254852433822744595, 3.23017139390104348321244125489, 4.06571360938822847675345244548, 4.75947880656934383239333674483, 6.03056347313571943101291809351, 7.32341442632051466377503374778, 8.138497803135048826467803154722, 8.640600675946470185780232884891, 9.180916448918465821904835909859, 9.956988459228173722299144400937

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