L-function 1-812-812.55-r1-0-0

• Label: 1-812-812.55-r1-0-0
• Degree: $1$
• Conductor: $812$
• Sign: $-0.995 + 0.0915i$
• 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

+ (−0.781 + 0.623i)3^-s + (−0.900 + 0.433i)5^-s + (0.222 − 0.974i)9^-s + (−0.974 + 0.222i)11^-s + (−0.222 − 0.974i)13^-s + (0.433 − 0.900i)15^-s − i·17^-s + (0.781 + 0.623i)19^-s + (0.900 + 0.433i)23^-s + (0.623 − 0.781i)25^-s + (0.433 + 0.900i)27^-s + (−0.433 − 0.900i)31^-s + (0.623 − 0.781i)33^-s + (0.974 + 0.222i)37^-s + (0.781 + 0.623i)39^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$812$    =    $2^{2} \cdot 7 \cdot 29$
Sign:	$-0.995 + 0.0915i$
Analytic conductor:	$87.2615$
Root analytic conductor:	$87.2615$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{812} (55, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 812,\ (1:\ ),\ -0.995 + 0.0915i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.01189793904 + 0.2593388985i$
$L(1)$	$\approx$	$0.6030885426 + 0.1295193452i$

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	$1 + (-0.781 + 0.623i)T$
	5	$1 + (-0.900 + 0.433i)T$
	11	$1 + (-0.974 + 0.222i)T$
	13	$1 + (-0.222 - 0.974i)T$
	17	$1 - iT$
	19	$1 + (0.781 + 0.623i)T$
	23	$1 + (0.900 + 0.433i)T$
	31	$1 + (-0.433 - 0.900i)T$
	37	$1 + (0.974 + 0.222i)T$

Imaginary part of the first few zeros on the critical line

−21.71621089365195604679499152760, −20.90110148115660027210178184313, −19.81730887235591497566009761018, −19.16214116434549495223776632890, −18.52780855344405570563737824894, −17.62201933986011042985582837842, −16.72448117065535928005557700193, −16.16288282479164051242936558187, −15.37216154218805239012284891937, −14.27049468897158860306826242470, −13.1498983443521795830913559804, −12.68772380885501107016915199676, −11.75062268274024621469867244886, −11.15187606369090883294235523978, −10.33158081146837545215552737714, −8.99175109559655239974952274103, −8.148467639677118404650507519945, −7.30973944395244817235670620514, −6.61204705808339150667778064284, −5.349559174737439058628723924732, −4.79406102411802986760500063431, −3.63604727293965786597545085588, −2.3157101287846803667257919856, −1.11261541151883082701787560129, −0.093903469645713564127230550629, 0.877648178308628961140018010830, 2.807227869143335964205996319211, 3.47629083181154333797197866062, 4.70447145390855243551588463479, 5.27303168299133114570770270934, 6.35915989929790730115746900850, 7.46624649420606383327471772937, 7.991790060057767886727127439426, 9.46219049330155213698681073179, 10.09812593769381545415578859287, 11.06671559767390903462635590743, 11.52657621869702904547195051808, 12.48774715752412486735223252707, 13.29086477692461430088922044552, 14.74023704365048221198103087590, 15.165803920260762023349653004291, 16.06177871981021402394213027697, 16.49427462959319244702230477410, 17.8017897607726494186460538985, 18.203386213390163237899503127941, 19.11178199757133025569816344333, 20.31979116630743055240360845098, 20.65985974181877157057556062176, 21.79334970674517111572957802465, 22.56982120430834923766344521660

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