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Properties

Label 1-847-847.678-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.427 - 0.904i$
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  + (−0.516 + 0.856i)2-s + (0.809 + 0.587i)3-s + (−0.466 − 0.884i)4-s + (−0.774 + 0.633i)5-s + (−0.921 + 0.389i)6-s + (0.998 + 0.0570i)8-s + (0.309 + 0.951i)9-s + (−0.142 − 0.989i)10-s + (0.142 − 0.989i)12-s + (−0.985 + 0.170i)13-s + (−0.998 + 0.0570i)15-s + (−0.564 + 0.825i)16-s + (0.993 − 0.113i)17-s + (−0.974 − 0.226i)18-s + (−0.870 + 0.491i)19-s + (0.921 + 0.389i)20-s + ⋯
L(s)  = 1  + (−0.516 + 0.856i)2-s + (0.809 + 0.587i)3-s + (−0.466 − 0.884i)4-s + (−0.774 + 0.633i)5-s + (−0.921 + 0.389i)6-s + (0.998 + 0.0570i)8-s + (0.309 + 0.951i)9-s + (−0.142 − 0.989i)10-s + (0.142 − 0.989i)12-s + (−0.985 + 0.170i)13-s + (−0.998 + 0.0570i)15-s + (−0.564 + 0.825i)16-s + (0.993 − 0.113i)17-s + (−0.974 − 0.226i)18-s + (−0.870 + 0.491i)19-s + (0.921 + 0.389i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.427 - 0.904i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (678, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.427 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2256164806 + 0.3561275181i\)
\(L(\frac12)\) \(\approx\) \(-0.2256164806 + 0.3561275181i\)
\(L(1)\) \(\approx\) \(0.5085434951 + 0.4928335698i\)
\(L(1)\) \(\approx\) \(0.5085434951 + 0.4928335698i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.516 + 0.856i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.774 + 0.633i)T \)
13 \( 1 + (-0.985 + 0.170i)T \)
17 \( 1 + (0.993 - 0.113i)T \)
19 \( 1 + (-0.870 + 0.491i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (-0.198 - 0.980i)T \)
31 \( 1 + (-0.897 + 0.441i)T \)
37 \( 1 + (0.696 + 0.717i)T \)
41 \( 1 + (0.0855 + 0.996i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (-0.974 + 0.226i)T \)
53 \( 1 + (-0.564 - 0.825i)T \)
59 \( 1 + (-0.0855 + 0.996i)T \)
61 \( 1 + (0.516 + 0.856i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (0.941 - 0.336i)T \)
79 \( 1 + (0.254 - 0.967i)T \)
83 \( 1 + (-0.0285 + 0.999i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (-0.774 - 0.633i)T \)
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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

−21.362359339078460906195331782007, −20.345365268127013711305763880298, −20.031156491562460148343538142417, −19.25462086248833390510592148391, −18.73073374083131712625676611560, −17.75182646694418367258386263873, −16.945835972967452690909590620650, −16.09740269046389666138471109983, −14.98103800767472203975440379268, −14.21174693998385371077706211189, −13.09361221864509013722659698072, −12.52724962888213355308176867046, −12.00105366459113111510410388960, −10.989817016954915934537310198133, −9.84742889170680509243207218130, −9.16767864979860433090947008891, −8.29744404213225314204790644397, −7.72457826417853833484773757177, −6.97392971545091234370306713160, −5.28816241567201929257305890654, −4.12248059558429858823602643385, −3.44824146159780610860378244590, −2.37532567574556384293116587237, −1.45780690297195114036174630272, −0.20077994840359182674379263578, 1.81255557701345139791991871374, 2.99529032964299230794884051762, 4.09331063174677466796791616263, 4.79964417305380555932423972162, 6.03969480629469996830033169557, 7.0681268405775548592993926848, 7.9425666265437083172811239891, 8.25087940069329079883255278646, 9.56862645619965370588385360318, 10.03153804630485278818994192504, 10.88849398899754871244875257877, 11.998899756346218058842093373691, 13.2331163111093980538751956196, 14.31747603785192511140783450947, 14.75224807106125775027973571588, 15.234875270943590212606067552849, 16.41950670327887615909973780218, 16.57274963561738543993440203692, 17.96659778653017859112880810342, 18.74021122337236431589354772255, 19.45462108483605095951310472285, 19.88205095185852378032687053155, 21.01884727194770733855082430316, 22.00720605320614439712291160460, 22.692615311168173597674444562220

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