L-function 1-847-847.237-r0-0-0

• Label: 1-847-847.237-r0-0-0
• Degree: $1$
• Conductor: $847$
• Sign: $-0.201 + 0.979i$
• Analytic cond.: $3.93345$
• Root an. cond.: $3.93345$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.466 + 0.884i)2^-s + (−0.309 − 0.951i)3^-s + (−0.564 + 0.825i)4^-s + (−0.198 + 0.980i)5^-s + (0.696 − 0.717i)6^-s + (−0.993 − 0.113i)8^-s + (−0.809 + 0.587i)9^-s + (−0.959 + 0.281i)10^-s + (0.959 + 0.281i)12^-s + (0.941 − 0.336i)13^-s + (0.993 − 0.113i)15^-s + (−0.362 − 0.931i)16^-s + (0.974 − 0.226i)17^-s + (−0.897 − 0.441i)18^-s + (0.516 − 0.856i)19^-s + (−0.696 − 0.717i)20^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$847$    =    $7 \cdot 11^{2}$
Sign:	$-0.201 + 0.979i$
Analytic conductor:	$3.93345$
Root analytic conductor:	$3.93345$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{847} (237, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 847,\ (0:\ ),\ -0.201 + 0.979i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8876781742 + 1.088465083i$
$L(1)$	$\approx$	$0.9949064722 + 0.5067025904i$

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	7	$1$
	11	$1$
good	2	$1 + (0.466 + 0.884i)T$
	3	$1 + (-0.309 - 0.951i)T$
	5	$1 + (-0.198 + 0.980i)T$
	13	$1 + (0.941 - 0.336i)T$
	17	$1 + (0.974 - 0.226i)T$
	19	$1 + (0.516 - 0.856i)T$
	23	$1 + (0.841 + 0.540i)T$
	29	$1 + (0.921 - 0.389i)T$
	31	$1 + (-0.610 + 0.791i)T$

Imaginary part of the first few zeros on the critical line

−21.651100459486630883841369800956, −20.96223307250466120584412405668, −20.63747417528147701359005896928, −19.81477625088605297169997378124, −18.86180118170564558724628563712, −17.99989920074875420147958146613, −16.86735529513651462996059886089, −16.30926847818778899341247806577, −15.4263326245475326837716391039, −14.55111195985018133284855165787, −13.75701259520439353847062395604, −12.73121341908050926914059586330, −12.05245387691542913846425958256, −11.324463257937921383347924942927, −10.46992772304103749459428470017, −9.68494176633832791363938613489, −8.91858178978711326507552304980, −8.17697634996180942219557350818, −6.38461351374075956059501118698, −5.42607150236073093266050303151, −4.9176884614453260771338601105, −3.81203952647562626115965212801, −3.405151786578308619869202325736, −1.79107984661021203050209379304, −0.67966814160027836238691425616, 1.14988941905933100342554779473, 2.87074309046720608349707006680, 3.33687849318026006108005519592, 4.84458767907547722283206816958, 5.73552856944427565930901890899, 6.541294171790754786610404024914, 7.16431848008505974054312456356, 7.912543087130799893974250692631, 8.72016268005483777741060397267, 10.03599157059721010599345598980, 11.252016526113570959874560478457, 11.779438244011117551915163184206, 12.83075482903470671422064271083, 13.57670733333999915322878248390, 14.161271320678326055932671805, 15.05449389974778252037754303857, 15.84176273694655784594936514390, 16.72341573820029367423123158375, 17.68339977943546207456824412866, 18.16800054542307402372230549243, 18.85498929142775639000916226023, 19.73893897176989004853440885637, 20.962702907948901035900546309704, 21.88448478952461123634803093502, 22.61321357029040799542021730477

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