L-function 1-812-812.471-r1-0-0

• Label: 1-812-812.471-r1-0-0
• Degree: $1$
• Conductor: $812$
• Sign: $0.986 - 0.164i$
• Analytic cond.: $87.2615$
• Root an. cond.: $87.2615$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.988 − 0.149i)3^-s + (0.826 + 0.563i)5^-s + (0.955 − 0.294i)9^-s + (−0.955 − 0.294i)11^-s + (−0.222 − 0.974i)13^-s + (0.900 + 0.433i)15^-s + (−0.5 + 0.866i)17^-s + (0.988 + 0.149i)19^-s + (−0.0747 − 0.997i)23^-s + (0.365 + 0.930i)25^-s + (0.900 − 0.433i)27^-s + (−0.0747 + 0.997i)31^-s + (−0.988 − 0.149i)33^-s + (0.955 − 0.294i)37^-s + (−0.365 − 0.930i)39^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$812$    =    $2^{2} \cdot 7 \cdot 29$
Sign:	$0.986 - 0.164i$
Analytic conductor:	$87.2615$
Root analytic conductor:	$87.2615$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{812} (471, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 812,\ (1:\ ),\ 0.986 - 0.164i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$3.815686899 - 0.3153273343i$
$L(1)$	$\approx$	$1.782927932 - 0.04002879594i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	7	$1$
	29	$1$
good	3	$1 + (0.988 - 0.149i)T$
	5	$1 + (0.826 + 0.563i)T$
	11	$1 + (-0.955 - 0.294i)T$
	13	$1 + (-0.222 - 0.974i)T$
	17	$1 + (-0.5 + 0.866i)T$
	19	$1 + (0.988 + 0.149i)T$
	23	$1 + (-0.0747 - 0.997i)T$
	31	$1 + (-0.0747 + 0.997i)T$
	37	$1 + (0.955 - 0.294i)T$

Imaginary part of the first few zeros on the critical line

−21.7387219794955739393808024947, −21.18601799687991920446585886416, −20.44991700867037090679966712100, −19.91172242607643535441366284896, −18.79320883193713631480253630960, −18.17059546342553725385722800948, −17.26847783508162608654983105284, −16.17338460506922734877666349271, −15.70844733862399296230265816842, −14.62740033691540477738503132109, −13.72982650449731059940004767991, −13.41358827664182169706726992585, −12.45533966432090341767458464103, −11.365871682260746452734588877341, −10.19667536288936351957369078509, −9.37556165630588193241867913572, −9.096408879223288378452666017931, −7.78996201536556454016409996189, −7.22036192653906767317845799424, −5.88249940056625273507635658660, −4.9084214948517474945848973165, −4.14717901091015222398201750191, −2.736756738552064314517301882579, −2.17018925762037607401353736480, −0.98419728198034136497471761204, 0.86982887477759512086616925398, 2.20473805920493138660151046512, 2.78039681827465718573036306141, 3.71713355073687006352515075913, 5.06376862100399764719012166134, 5.99177515125734637789089421564, 6.99941827740043358557931734332, 7.8775330301674720830300889699, 8.61169445087740026246546922368, 9.677187074467008730047241095073, 10.310656538269740925199776738796, 11.055062260200080920372071406592, 12.70788036310199459952421778677, 12.96673812690162136892689966961, 14.02521101496631233450015357568, 14.53350407519906265048472199098, 15.43007214814175853281249686776, 16.15131418682758901815332834964, 17.47335062510647449978649974985, 18.12077261284394651373789695864, 18.71098729032911686539177183450, 19.68219869570045226576762235206, 20.41874532690102577270195298059, 21.16771277883806982354203177202, 21.85862127876517959362285551718

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