L-function 1-35-35.2-r1-0-0

• Label: 1-35-35.2-r1-0-0
• Degree: $1$
• Conductor: $35$
• Sign: $0.908 - 0.417i$
• Analytic cond.: $3.76127$
• Root an. cond.: $3.76127$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.866 − 0.5i)2^-s + (0.866 + 0.5i)3^-s + (0.5 − 0.866i)4^-s + 6^-s − i·8^-s + (0.5 + 0.866i)9^-s + (−0.5 + 0.866i)11^-s + (0.866 − 0.5i)12^-s − i·13^-s + (−0.5 − 0.866i)16^-s + (−0.866 − 0.5i)17^-s + (0.866 + 0.5i)18^-s + (0.5 + 0.866i)19^-s + i·22^-s + (−0.866 + 0.5i)23^-s + (0.5 − 0.866i)24^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$35$    =    $5 \cdot 7$
Sign:	$0.908 - 0.417i$
Analytic conductor:	$3.76127$
Root analytic conductor:	$3.76127$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{35} (2, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 35,\ (1:\ ),\ 0.908 - 0.417i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.718111101 - 0.5939212068i$
$L(1)$	$\approx$	$2.022687173 - 0.3439784556i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	7	$1$
good	2	$1 + (0.866 - 0.5i)T$
	3	$1 + (0.866 + 0.5i)T$
	11	$1 + (-0.5 + 0.866i)T$
	13	$1 - iT$
	17	$1 + (-0.866 - 0.5i)T$
	19	$1 + (0.5 + 0.866i)T$
	23	$1 + (-0.866 + 0.5i)T$
	29	$1 - T$
	31	$1 + (-0.5 + 0.866i)T$

Imaginary part of the first few zeros on the critical line

−35.59910556888870191104038261084, −34.5491197196213397853178449292, −33.139790922880960360837633984477, −31.98429806201945131277114662520, −31.14316437994159622782128062278, −30.12097472286172158098414098320, −28.91438142656687972759492667454, −26.53942157582756157533475900835, −25.94738826735708365552515402428, −24.31453507290520542980915996304, −23.96053750711599078301222154631, −22.10785158455013824657689698595, −20.9722004665645462449841208562, −19.6913951972389108501187299340, −18.16097632199931193431851527847, −16.40798799212429862133596127345, −15.09673006095517454118607126277, −13.870216007044681308131658372130, −12.99469199329751140900192168461, −11.40305618408171642726960277184, −8.9326723563768611725471447826, −7.59617739475376023513821328415, −6.195058614860654066255917608380, −4.10981455216769926587429169531, −2.45955293864876159220403598747, 2.25482763699762490408729632759, 3.800184639831668899720305117076, 5.30810329549512608468832877963, 7.53940282731089718088345180747, 9.56053378114159156516744300472, 10.71454759968517399891237260839, 12.54940421952370673502148455321, 13.75107163835168794846892456630, 15.003608723005802107184334105923, 15.93436159450332709011586603405, 18.302937061750072741358329909129, 19.954979484248753356949264425323, 20.51981840154720851774899142914, 21.867376268165700036905272143251, 22.96795193488978953650872254987, 24.567079380133396770061055546504, 25.59616376233569921601209715488, 27.188152775860030134872747315948, 28.40474534221674878824998352695, 29.879416518472621948159014955233, 31.00485230188198038257288282362, 31.80334912458809387038562252908, 32.96917005755833550252605032144, 33.84273960730530060122891935750, 35.89377509879203932391478691188

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