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Properties

Label 1-847-847.87-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.968 + 0.247i$
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  + (−0.235 + 0.971i)2-s + (0.5 + 0.866i)3-s + (−0.888 − 0.458i)4-s + (−0.981 + 0.189i)5-s + (−0.959 + 0.281i)6-s + (0.654 − 0.755i)8-s + (−0.5 + 0.866i)9-s + (0.0475 − 0.998i)10-s + (−0.0475 − 0.998i)12-s + (0.841 − 0.540i)13-s + (−0.654 − 0.755i)15-s + (0.580 + 0.814i)16-s + (0.928 − 0.371i)17-s + (−0.723 − 0.690i)18-s + (0.928 + 0.371i)19-s + (0.959 + 0.281i)20-s + ⋯
L(s)  = 1  + (−0.235 + 0.971i)2-s + (0.5 + 0.866i)3-s + (−0.888 − 0.458i)4-s + (−0.981 + 0.189i)5-s + (−0.959 + 0.281i)6-s + (0.654 − 0.755i)8-s + (−0.5 + 0.866i)9-s + (0.0475 − 0.998i)10-s + (−0.0475 − 0.998i)12-s + (0.841 − 0.540i)13-s + (−0.654 − 0.755i)15-s + (0.580 + 0.814i)16-s + (0.928 − 0.371i)17-s + (−0.723 − 0.690i)18-s + (0.928 + 0.371i)19-s + (0.959 + 0.281i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.968 + 0.247i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.968 + 0.247i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1405420551 + 1.119735115i\)
\(L(\frac12)\) \(\approx\) \(0.1405420551 + 1.119735115i\)
\(L(1)\) \(\approx\) \(0.6164795771 + 0.6858466101i\)
\(L(1)\) \(\approx\) \(0.6164795771 + 0.6858466101i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.235 + 0.971i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.981 + 0.189i)T \)
13 \( 1 + (0.841 - 0.540i)T \)
17 \( 1 + (0.928 - 0.371i)T \)
19 \( 1 + (0.928 + 0.371i)T \)
23 \( 1 + (0.580 + 0.814i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (-0.0475 + 0.998i)T \)
37 \( 1 + (-0.888 + 0.458i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (-0.723 + 0.690i)T \)
53 \( 1 + (0.580 - 0.814i)T \)
59 \( 1 + (-0.235 - 0.971i)T \)
61 \( 1 + (0.723 - 0.690i)T \)
67 \( 1 + (0.723 + 0.690i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.995 + 0.0950i)T \)
79 \( 1 + (-0.981 + 0.189i)T \)
83 \( 1 + (0.415 + 0.909i)T \)
89 \( 1 + (0.786 + 0.618i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
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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

−21.4447671072735802027113576772, −20.62923649835827977697604287738, −20.18660088444656065073729498774, −19.13739875499014484243682741424, −18.931388747343735837581157892537, −18.134824610430986843515652534839, −17.137712957505441922358922266019, −16.28453743677792739763950751778, −15.1252297551872804948204846517, −14.232512185785639416025269651552, −13.41741068992944897676680686942, −12.72030233463496502203801051366, −11.81903594247666275397273183982, −11.48635757854660061573085099103, −10.324689743972783804175450621399, −9.185588846652377685922860212100, −8.53642014686661805598433409041, −7.81412047057786647296278260940, −7.02203504744079511975733860664, −5.68045755231290706140329577800, −4.34934921363181335477377628693, −3.551611189524881190626994903720, −2.7576289085352735189164240224, −1.53949803580234032730727373222, −0.64519914292188492390494674911, 1.19543811644695907395400367459, 3.354000554923987096438649533479, 3.5450917467704746984514002322, 4.94062123532019379444270326054, 5.417457056793494478141290764507, 6.81153283047899158973355196072, 7.68498277684332205990140464393, 8.32684556753323231383478070277, 9.077409290951703658018142730826, 10.051989195394727248837345674547, 10.75583865982827670385590914221, 11.78535090082237716164112212705, 12.99604052912008294819512295596, 14.03024677924571578816081323458, 14.5515150302794676300747906824, 15.488849034088031291142022660804, 15.92340552895240743102212202988, 16.49817365671898975787170637428, 17.55950600744159294034340046046, 18.5509515102838654204873092372, 19.179613328698681633308590320757, 20.058660209152712817170526724537, 20.77985474234068938994235933525, 21.87139839296364846853566561565, 22.69108325881476144839116399857

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