↑

Properties

Label 1-847-847.67-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.333 - 0.942i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0475 + 0.998i)2-s + (−0.5 − 0.866i)3-s + (−0.995 + 0.0950i)4-s + (−0.786 − 0.618i)5-s + (0.841 − 0.540i)6-s + (−0.142 − 0.989i)8-s + (−0.5 + 0.866i)9-s + (0.580 − 0.814i)10-s + (0.580 + 0.814i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)15-s + (0.981 − 0.189i)16-s + (0.235 + 0.971i)17-s + (−0.888 − 0.458i)18-s + (0.235 − 0.971i)19-s + (0.841 + 0.540i)20-s + ⋯
L(s)  = 1  + (0.0475 + 0.998i)2-s + (−0.5 − 0.866i)3-s + (−0.995 + 0.0950i)4-s + (−0.786 − 0.618i)5-s + (0.841 − 0.540i)6-s + (−0.142 − 0.989i)8-s + (−0.5 + 0.866i)9-s + (0.580 − 0.814i)10-s + (0.580 + 0.814i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)15-s + (0.981 − 0.189i)16-s + (0.235 + 0.971i)17-s + (−0.888 − 0.458i)18-s + (0.235 − 0.971i)19-s + (0.841 + 0.540i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.333 - 0.942i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.333 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3001480201 - 0.4246909767i\)
\(L(\frac12)\) \(\approx\) \(0.3001480201 - 0.4246909767i\)
\(L(1)\) \(\approx\) \(0.6569176385 + 0.02520116390i\)
\(L(1)\) \(\approx\) \(0.6569176385 + 0.02520116390i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0475 + 0.998i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.786 - 0.618i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (0.235 + 0.971i)T \)
19 \( 1 + (0.235 - 0.971i)T \)
23 \( 1 + (0.981 - 0.189i)T \)
29 \( 1 + (-0.959 + 0.281i)T \)
31 \( 1 + (0.580 - 0.814i)T \)
37 \( 1 + (-0.995 - 0.0950i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.888 + 0.458i)T \)
53 \( 1 + (0.981 + 0.189i)T \)
59 \( 1 + (0.0475 - 0.998i)T \)
61 \( 1 + (-0.888 + 0.458i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.327 + 0.945i)T \)
79 \( 1 + (-0.786 - 0.618i)T \)
83 \( 1 + (-0.654 + 0.755i)T \)
89 \( 1 + (0.723 - 0.690i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.44050024536591243493367629577, −21.31312238177783111058097321168, −21.00997937120222181431863613991, −20.06178034535370803415495924802, −19.1981307517954990625948757609, −18.5051922512607522175325071341, −17.784561599034982589156018466236, −16.65159482541905327179561961663, −16.053827485937753108285783909596, −14.931377787420706929371268480781, −14.39208929882443080169268869545, −13.40699044093893214623459959986, −12.09325133789368978095040735299, −11.69434716698290144437161010146, −10.98546596593648081501994765398, −10.25189435551390525338174879116, −9.4046842711921454468957867334, −8.62516954134859183286248995480, −7.4359253183058691750162972004, −6.26589850688627021370656620918, −5.15458704414652252750792622975, −4.333446390555032952745848252977, −3.53746097785779186672023625492, −2.84034590561480220790765067369, −1.26833861536455112643575193578, 0.29947661537361698398988596973, 1.33168802190806493755863346674, 3.11809360014225626075594370120, 4.25202005485664938273259762366, 5.24432776589860395255010853292, 5.85228404118734854815556619716, 6.96401486171194438860294249736, 7.59245303283145635624931401479, 8.38090568753433702151528407619, 9.025586194255743141200363482593, 10.46480486783971928157569033374, 11.38187903991793497421312425192, 12.43755871561315624168984754928, 12.97093045648443329024095983370, 13.59450948241017058375207861007, 14.83148773962874606797651646656, 15.49036640575719598540156858736, 16.31358669477821472010000341102, 17.15736762907709472854797449880, 17.56657437344951877644906008736, 18.65327993019612026252957955938, 19.197299766596304808973808533870, 20.06618667412070027186809160940, 21.16578106248998289735072683094, 22.3812569230786274502220515969

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