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Properties

Label 1-847-847.216-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.413 + 0.910i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.897 + 0.441i)2-s + (0.809 + 0.587i)3-s + (0.610 − 0.791i)4-s + (0.736 + 0.676i)5-s + (−0.985 − 0.170i)6-s + (−0.198 + 0.980i)8-s + (0.309 + 0.951i)9-s + (−0.959 − 0.281i)10-s + (0.959 − 0.281i)12-s + (−0.564 − 0.825i)13-s + (0.198 + 0.980i)15-s + (−0.254 − 0.967i)16-s + (−0.921 + 0.389i)17-s + (−0.696 − 0.717i)18-s + (0.974 − 0.226i)19-s + (0.985 − 0.170i)20-s + ⋯
L(s)  = 1  + (−0.897 + 0.441i)2-s + (0.809 + 0.587i)3-s + (0.610 − 0.791i)4-s + (0.736 + 0.676i)5-s + (−0.985 − 0.170i)6-s + (−0.198 + 0.980i)8-s + (0.309 + 0.951i)9-s + (−0.959 − 0.281i)10-s + (0.959 − 0.281i)12-s + (−0.564 − 0.825i)13-s + (0.198 + 0.980i)15-s + (−0.254 − 0.967i)16-s + (−0.921 + 0.389i)17-s + (−0.696 − 0.717i)18-s + (0.974 − 0.226i)19-s + (0.985 − 0.170i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.413 + 0.910i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (216, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.413 + 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7613503162 + 1.181309593i\)
\(L(\frac12)\) \(\approx\) \(0.7613503162 + 1.181309593i\)
\(L(1)\) \(\approx\) \(0.8928598248 + 0.5542174832i\)
\(L(1)\) \(\approx\) \(0.8928598248 + 0.5542174832i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.897 + 0.441i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.736 + 0.676i)T \)
13 \( 1 + (-0.564 - 0.825i)T \)
17 \( 1 + (-0.921 + 0.389i)T \)
19 \( 1 + (0.974 - 0.226i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (-0.0855 + 0.996i)T \)
31 \( 1 + (0.0285 + 0.999i)T \)
37 \( 1 + (0.941 - 0.336i)T \)
41 \( 1 + (-0.466 + 0.884i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (-0.696 + 0.717i)T \)
53 \( 1 + (-0.254 + 0.967i)T \)
59 \( 1 + (0.466 + 0.884i)T \)
61 \( 1 + (0.897 + 0.441i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.516 + 0.856i)T \)
73 \( 1 + (-0.362 + 0.931i)T \)
79 \( 1 + (-0.993 - 0.113i)T \)
83 \( 1 + (0.774 - 0.633i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.736 - 0.676i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.4986161296044183116739161923, −20.77141721790880152793752699071, −20.24658431460411781939264345478, −19.45103001469757533224001618473, −18.74790093407577847205848850519, −17.928739911228633735092551445472, −17.30260425511433950548445081745, −16.473822501891999082105560768447, −15.55357297437707356373857475041, −14.516069280159500524827809148184, −13.39043531474385924628934551980, −13.11506885584839901325861189855, −11.95093181732731908322773528012, −11.40146461016686422649996140568, −9.81007028457427400586300959390, −9.549815485857331263766465050098, −8.75193231783472760536841808655, −7.91306181458874100393485700005, −7.0457136885687005451103158503, −6.237621681388888298417357475537, −4.78358518462376494477825720766, −3.59883771777660715013326009250, −2.43621237552869787943574581537, −1.86882882209429709840511733563, −0.782135534362566321531468334716, 1.42381714457957449109169052693, 2.54964181749641173386800751642, 3.138737666884331271112735114840, 4.79033702973574379042986307840, 5.58418540194466952922005848801, 6.774990514110685651286701188819, 7.41654807448700020087396815773, 8.45905853646262680744136439672, 9.177166835008721778833399871664, 9.94736826313661125473051283211, 10.57199379667098417362257302446, 11.26025255971606591904595088061, 12.83553408809858995799963810345, 13.776264949396714982511834984331, 14.62104507214029502917896916488, 15.06888133601534685016760450325, 15.910722614585951411113706184729, 16.77477790122431979261611581564, 17.67927368189013581723855029556, 18.26797984850029910114167972455, 19.16350470054297900032473666890, 19.98602979142828060635816196161, 20.462300873409193390564176603573, 21.57564385589329981206805297536, 22.160634732230995954941773299474

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