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Properties

Label 1-847-847.331-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.821 + 0.570i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.821 + 0.570i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.821 + 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4780491391 + 1.525521014i\)
\(L(\frac12)\) \(\approx\) \(0.4780491391 + 1.525521014i\)
\(L(1)\) \(\approx\) \(0.7899629878 + 0.8764047871i\)
\(L(1)\) \(\approx\) \(0.7899629878 + 0.8764047871i\)

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

−21.80560665416619146935370582671, −20.85614193087673352152453318602, −20.41638671807291836532075395549, −19.284797884225088117899498453945, −18.561566717384574740113342338690, −18.03142841970158208488378135142, −17.27698592407504918359537620392, −16.50440467663097373124788239658, −15.1229868950035677458283884628, −13.98287133623139346261184691556, −13.4891721369818217672160030147, −12.95481394333561345256099455696, −11.93229792742372100017265242548, −11.407411080881018683659740153878, −10.31677177901795353219649926642, −9.730025313765967422163456153959, −8.63631116791784099531974426029, −7.725651445972593993596370579012, −6.43218709975640937369568709046, −5.530989753881462819720950809144, −5.172192063549953493498492083176, −3.59079892424076257065044483990, −2.62525825821664004557635487405, −1.57818684519549244761177157654, −0.89995330294471310082209903726, 1.19591668198278429028502203858, 3.04452345283579911300001612331, 3.82881124595508823071385086321, 5.072577755036440733629473076485, 5.471012669626734703801216416552, 6.44251302010622166814235606922, 7.08429465965936676760620548209, 8.6236373392673057419148385017, 9.0905005282058698014384031136, 10.0628750174158139529278293739, 10.73965412751118240978568893943, 11.958660268271974743622307806521, 12.82790748492372591357621564437, 13.93866897893441355430083203142, 14.333982443627825967282647249999, 15.28649626050778391863182438086, 16.11784853718836426050053365270, 16.7142372303150880429105130876, 17.40657068359200508304045584439, 18.17202403812510487315086338321, 18.84047277651315330338623369479, 20.47236475461317992541043797562, 21.104556026869001943294854080449, 21.84745809229349241643681023604, 22.34272507451611280927264649371

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