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Properties

Label 1-847-847.167-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.0622 + 0.998i$
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  + (−0.610 − 0.791i)2-s + (−0.309 + 0.951i)3-s + (−0.254 + 0.967i)4-s + (−0.0855 + 0.996i)5-s + (0.941 − 0.336i)6-s + (0.921 − 0.389i)8-s + (−0.809 − 0.587i)9-s + (0.841 − 0.540i)10-s + (−0.841 − 0.540i)12-s + (−0.362 − 0.931i)13-s + (−0.921 − 0.389i)15-s + (−0.870 − 0.491i)16-s + (0.696 + 0.717i)17-s + (0.0285 + 0.999i)18-s + (0.897 + 0.441i)19-s + (−0.941 − 0.336i)20-s + ⋯
L(s)  = 1  + (−0.610 − 0.791i)2-s + (−0.309 + 0.951i)3-s + (−0.254 + 0.967i)4-s + (−0.0855 + 0.996i)5-s + (0.941 − 0.336i)6-s + (0.921 − 0.389i)8-s + (−0.809 − 0.587i)9-s + (0.841 − 0.540i)10-s + (−0.841 − 0.540i)12-s + (−0.362 − 0.931i)13-s + (−0.921 − 0.389i)15-s + (−0.870 − 0.491i)16-s + (0.696 + 0.717i)17-s + (0.0285 + 0.999i)18-s + (0.897 + 0.441i)19-s + (−0.941 − 0.336i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0622 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0622 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.0622 + 0.998i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.0622 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5525239305 + 0.5880721776i\)
\(L(\frac12)\) \(\approx\) \(0.5525239305 + 0.5880721776i\)
\(L(1)\) \(\approx\) \(0.6785782395 + 0.1817544322i\)
\(L(1)\) \(\approx\) \(0.6785782395 + 0.1817544322i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.610 - 0.791i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.0855 + 0.996i)T \)
13 \( 1 + (-0.362 - 0.931i)T \)
17 \( 1 + (0.696 + 0.717i)T \)
19 \( 1 + (0.897 + 0.441i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (0.985 - 0.170i)T \)
31 \( 1 + (0.998 + 0.0570i)T \)
37 \( 1 + (0.774 + 0.633i)T \)
41 \( 1 + (-0.564 + 0.825i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.0285 - 0.999i)T \)
53 \( 1 + (-0.870 + 0.491i)T \)
59 \( 1 + (0.564 + 0.825i)T \)
61 \( 1 + (0.610 - 0.791i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.466 - 0.884i)T \)
73 \( 1 + (-0.736 + 0.676i)T \)
79 \( 1 + (-0.974 + 0.226i)T \)
83 \( 1 + (0.198 + 0.980i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (-0.0855 - 0.996i)T \)
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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

−22.2049908177800520817163890404, −20.850893044259348499366483785258, −20.08704661497259104668015648592, −19.19072650737103617236513195871, −18.76134002138071965155093826119, −17.6654776235420092017067547497, −17.249859321613327987866667018511, −16.26838748820838614420462539187, −15.9761579186006476708505326102, −14.488160465185564569088584266628, −13.928350013544238580556667799241, −13.077242064290246045885132242102, −12.077808693552327618516948571946, −11.468351931748085678633399116766, −10.211635219883142487987062850877, −9.219400562077423456297599978747, −8.5844785454165629358966888389, −7.65519749377516687037218746778, −7.021850669050241837443077273106, −6.08051825220511577285380824427, −5.1737002206745806012232951843, −4.510107488625817652273834089466, −2.58787903728529389353318827193, −1.38401014193051143828009591278, −0.58068221214414030827308547135, 1.156987049611813738358369299322, 2.827663026347905463803505046961, 3.21179433257591178727564764758, 4.213117584693962914911220384625, 5.34075207198595297145281679554, 6.40429203910715441231168113444, 7.64196306859432938624078772616, 8.28891601899651765289962060995, 9.6074209952882654775722954181, 10.06194105125511104991099851752, 10.67725089367879609202189742308, 11.5742179256574795724142735420, 12.11838775498068976927446977444, 13.3419792127064190366643580203, 14.34261006040601228764565420807, 15.19237275486000736950288504791, 15.94229228706552978779176997495, 16.93269021520476997968632865018, 17.6164663508908112531365176721, 18.25465353662124400071479415735, 19.26544072571047944355212001662, 19.88300152786486950059372786046, 20.79329620987230688388281490838, 21.52922620017957328582705035264, 22.10008060179430104168670617952

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