L-function 1-812-812.391-r1-0-0

• Label: 1-812-812.391-r1-0-0
• Degree: $1$
• Conductor: $812$
• Sign: $0.965 + 0.259i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$812$    =    $2^{2} \cdot 7 \cdot 29$
Sign:	$0.965 + 0.259i$
Analytic conductor:	$87.2615$
Root analytic conductor:	$87.2615$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{812} (391, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 812,\ (1:\ ),\ 0.965 + 0.259i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.630135477 + 0.2154626732i$
$L(1)$	$\approx$	$0.9517013155 + 0.1125079299i$

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

Imaginary part of the first few zeros on the critical line

−22.06373270275110158215859425656, −21.50768423163882883011852780701, −20.085418132290635607191237349684, −19.45700869126552828798916148233, −18.65295085115692363420027029819, −18.19723910011048586381359105988, −17.30631796676732836276863951771, −16.42483939510701565516799618612, −15.61465102541750096889534751825, −14.436461793834737673431615218808, −13.8401106721862322508649432871, −13.200446837023743422030781150120, −11.8065315575150958921966069630, −11.49173918967544873029126655650, −10.82903393262953512803218288949, −9.49981341588200535103852218835, −8.63052848714155998272674339019, −7.4220283957171582704277513750, −6.93669312462765441070534037066, −6.15435656219510719410677503908, −5.20191614023857814348396612416, −3.811739396806747443364326330948, −2.895546728958327065127181332202, −1.78824100785608635936829710988, −0.66277461938204213484040665615, 0.640761363806244289401308861897, 1.737459500603156902096997814080, 3.51416723036408021611930144244, 4.030442097774613574119199487224, 5.07063064338541377597921173049, 5.74945148099635242897293173170, 6.779812453157478333523251943283, 8.19610417345402361098239993320, 8.753730355483468377529513756114, 9.72128118014095509238769344812, 10.44255049226571541503059308743, 11.37405786684351988856078589735, 12.32407242843906342773967149447, 12.76112609130609427775531299476, 14.10547061196898733580185734609, 14.9639395124007628628815807153, 15.67614354128674981451963602598, 16.467020243205754984008123161215, 17.100404999514990534994772822278, 17.748837843124435759106281932461, 18.89950215808269555013194782935, 20.066479488649052735621867963456, 20.44505138894909160048816166017, 21.1369265451978418027215415112, 22.157576563497201904882402886895

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