↑

Properties

Label 1-847-847.214-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.770 - 0.636i$
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  + (0.861 − 0.508i)2-s + (0.669 − 0.743i)3-s + (0.483 − 0.875i)4-s + (−0.761 + 0.647i)5-s + (0.198 − 0.980i)6-s + (−0.0285 − 0.999i)8-s + (−0.104 − 0.994i)9-s + (−0.327 + 0.945i)10-s + (−0.327 − 0.945i)12-s + (0.0855 − 0.996i)13-s + (−0.0285 + 0.999i)15-s + (−0.532 − 0.846i)16-s + (0.449 + 0.893i)17-s + (−0.595 − 0.803i)18-s + (−0.710 − 0.703i)19-s + (0.198 + 0.980i)20-s + ⋯
L(s)  = 1  + (0.861 − 0.508i)2-s + (0.669 − 0.743i)3-s + (0.483 − 0.875i)4-s + (−0.761 + 0.647i)5-s + (0.198 − 0.980i)6-s + (−0.0285 − 0.999i)8-s + (−0.104 − 0.994i)9-s + (−0.327 + 0.945i)10-s + (−0.327 − 0.945i)12-s + (0.0855 − 0.996i)13-s + (−0.0285 + 0.999i)15-s + (−0.532 − 0.846i)16-s + (0.449 + 0.893i)17-s + (−0.595 − 0.803i)18-s + (−0.710 − 0.703i)19-s + (0.198 + 0.980i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.770 - 0.636i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.770 - 0.636i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8305147378 - 2.309438808i\)
\(L(\frac12)\) \(\approx\) \(0.8305147378 - 2.309438808i\)
\(L(1)\) \(\approx\) \(1.389191676 - 1.130038604i\)
\(L(1)\) \(\approx\) \(1.389191676 - 1.130038604i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.861 - 0.508i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (-0.761 + 0.647i)T \)
13 \( 1 + (0.0855 - 0.996i)T \)
17 \( 1 + (0.449 + 0.893i)T \)
19 \( 1 + (-0.710 - 0.703i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (0.774 - 0.633i)T \)
31 \( 1 + (-0.290 - 0.956i)T \)
37 \( 1 + (0.123 - 0.992i)T \)
41 \( 1 + (-0.736 + 0.676i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (-0.595 + 0.803i)T \)
53 \( 1 + (-0.532 + 0.846i)T \)
59 \( 1 + (-0.217 + 0.976i)T \)
61 \( 1 + (0.00951 - 0.999i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (-0.362 - 0.931i)T \)
73 \( 1 + (0.345 + 0.938i)T \)
79 \( 1 + (0.380 + 0.924i)T \)
83 \( 1 + (0.696 - 0.717i)T \)
89 \( 1 + (0.0475 + 0.998i)T \)
97 \( 1 + (0.941 - 0.336i)T \)
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

−22.43724462453555282926478658220, −21.515328232051802595012775720704, −20.926612633363361144580606282364, −20.339166089873725048655552654415, −19.4856644777482786751874251268, −18.62692681176673471812266455476, −17.10593911748815496585434696941, −16.40036203124140192206607182509, −16.06173513312741784742872384300, −15.08239054707361827537623670681, −14.4948336323566133485781776175, −13.70219356160543350905761062105, −12.82617749598589805696888496969, −11.93467185233053567100282820850, −11.22456938524851636474975863943, −10.08660327897848792996812241987, −8.81040386685444684195686694833, −8.47147544540717070673351627215, −7.42671544513378578639525675747, −6.58475354109645281876175253381, −5.04581397739830998637101066292, −4.78734009755399668109989248560, −3.72450948277742360282159200444, −3.10385743896823714668782811174, −1.77472805635873335614093881153, 0.74441156386142306131099886649, 2.03891756852137240478608644950, 2.99241956176786802082851902974, 3.55043360022137328816977156681, 4.59118452666831371114773247408, 5.92851062035702861932668458365, 6.65279348285204604191582850199, 7.5709571514566703151056392005, 8.32321997681979745693500909029, 9.560280167221036060442591679280, 10.60203623518867440999464854744, 11.28833125739613640777612252619, 12.2343773611300785570605835508, 12.88325601305515250022313235783, 13.57193897846804168170950051468, 14.578975829915260627712498745660, 15.11806780602301308118789807698, 15.60432817499503367951819065213, 17.092167902300240664412177590306, 18.17846365689918129674038352082, 18.91794750431086553787298223668, 19.56663950807069945203039952654, 20.01857117396222086487930454375, 21.00144955868680666205171966828, 21.74945706360448068303466295241

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