L-function 1-403-403.178-r1-0-0

• Label: 1-403-403.178-r1-0-0
• Degree: $1$
• Conductor: $403$
• Sign: $-0.979 - 0.201i$
• Analytic cond.: $43.3083$
• Root an. cond.: $43.3083$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.104 + 0.994i)2^-s + (−0.913 − 0.406i)3^-s + (−0.978 − 0.207i)4^-s + 5^-s + (0.5 − 0.866i)6^-s + (−0.978 − 0.207i)7^-s + (0.309 − 0.951i)8^-s + (0.669 + 0.743i)9^-s + (−0.104 + 0.994i)10^-s + (−0.669 + 0.743i)11^-s + (0.809 + 0.587i)12^-s + (0.309 − 0.951i)14^-s + (−0.913 − 0.406i)15^-s + (0.913 + 0.406i)16^-s + (−0.669 − 0.743i)17^-s + (−0.809 + 0.587i)18^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$403$    =    $13 \cdot 31$
Sign:	$-0.979 - 0.201i$
Analytic conductor:	$43.3083$
Root analytic conductor:	$43.3083$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{403} (178, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 403,\ (1:\ ),\ -0.979 - 0.201i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.03124287472 + 0.3070076447i$
$L(1)$	$\approx$	$0.5959907302 + 0.2324691559i$

Euler product

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

	$p$	$F_p(T)$
bad	13	$1$
	31	$1$
good	2	$1 + (-0.104 + 0.994i)T$
	3	$1 + (-0.913 - 0.406i)T$
	5	$1 + T$
	7	$1 + (-0.978 - 0.207i)T$
	11	$1 + (-0.669 + 0.743i)T$
	17	$1 + (-0.669 - 0.743i)T$
	19	$1 + (0.913 - 0.406i)T$
	23	$1 + (0.978 - 0.207i)T$
	29	$1 + (0.104 - 0.994i)T$

Imaginary part of the first few zeros on the critical line

−23.33008327754312509962339105906, −22.49336614133212433913430252320, −21.841990480314969674514704357779, −21.31645472294812466824322723272, −20.40737763952172023651794718947, −19.241924754816633096678284840202, −18.37789758403356389938895569533, −17.74515828768417873169262601136, −16.77523073390281856538583070712, −16.076524059850168413263838395928, −14.74618141364820622903516733695, −13.393614696974314750499006184741, −12.98570869935245494050490220815, −11.99174248497260522651819795018, −10.87389426309414147322233869729, −10.31924744798611734638725985495, −9.47620901018364957608607690836, −8.69469836874782113447392158774, −6.90421478098786356905532452132, −5.73062015322129168215129252131, −5.19158352497756472730686225829, −3.739849394687899799358038533121, −2.79481596241504686818159960394, −1.382632805470570639198332168036, −0.11662047496213342066576269023, 1.11916401882380562227293641693, 2.760434460774902368482674136655, 4.657780920048784338528417660, 5.28412659525794027190793181081, 6.43387317834153635405352521057, 6.82240656691398790550061405258, 7.89996830407025391701181342844, 9.51433636549621085687920151650, 9.81927573834122808523850169247, 11.01065177100901060078653989753, 12.48901955561422422790108985321, 13.28734573995665854817722271274, 13.66806454197636203455517613545, 15.15122624651895233622541031119, 16.01252209885314902219761206329, 16.76927906136454112568778819272, 17.54613606635941928073127394841, 18.202915878002825768335005250949, 18.90217806089174638469667345546, 20.201141052037175835477552558528, 21.53244381641270075405710088661, 22.411533776123812597005605938781, 22.872209160968902184304192572064, 23.70682000153099927165354104481, 24.8160934920953562671123349750

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