L-function 1-319-319.169-r0-0-0

• Label: 1-319-319.169-r0-0-0
• Degree: $1$
• Conductor: $319$
• Sign: $0.944 + 0.329i$
• Analytic cond.: $1.48142$
• Root an. cond.: $1.48142$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.995 + 0.0896i)2^-s + (0.983 + 0.178i)3^-s + (0.983 − 0.178i)4^-s + (0.858 + 0.512i)5^-s + (−0.995 − 0.0896i)6^-s + (0.473 − 0.880i)7^-s + (−0.963 + 0.266i)8^-s + (0.936 + 0.351i)9^-s + (−0.900 − 0.433i)10^-s + 12^-s + (−0.550 + 0.834i)13^-s + (−0.393 + 0.919i)14^-s + (0.753 + 0.657i)15^-s + (0.936 − 0.351i)16^-s + (0.309 − 0.951i)17^-s + (−0.963 − 0.266i)18^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$319$    =    $11 \cdot 29$
Sign:	$0.944 + 0.329i$
Analytic conductor:	$1.48142$
Root analytic conductor:	$1.48142$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{319} (169, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 319,\ (0:\ ),\ 0.944 + 0.329i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.396700481 + 0.2365383534i$
$L(1)$	$\approx$	$1.149867152 + 0.1305880229i$

Euler product

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

	$p$	$F_p(T)$
bad	11	$1$
	29	$1$
good	2	$1 + (-0.995 + 0.0896i)T$
	3	$1 + (0.983 + 0.178i)T$
	5	$1 + (0.858 + 0.512i)T$
	7	$1 + (0.473 - 0.880i)T$
	13	$1 + (-0.550 + 0.834i)T$
	17	$1 + (0.309 - 0.951i)T$
	19	$1 + (0.473 + 0.880i)T$
	23	$1 + (-0.222 - 0.974i)T$
	31	$1 + (-0.995 + 0.0896i)T$

Imaginary part of the first few zeros on the critical line

−25.3016217590687438044692002791, −24.55187755032580899137878666125, −23.9459391865250526973600517298, −21.765735641994392836775319469215, −21.47997005678744848862755112728, −20.37686154122825209456431024563, −19.79613229574075910696844761804, −18.79678840967849339426768925969, −17.87874753504573115518395852629, −17.375002241748960251949422288570, −16.01561507009324372277235619431, −15.17726774871349899441512686424, −14.35542897927129127473092806788, −12.98633567791526493000406326616, −12.37193035588176727213149548529, −11.0206649497037316016009999556, −9.75763597260754743398852804772, −9.28078603605351988052463695768, −8.31153918707000889593172860393, −7.61689104540458570394770028626, −6.23580726772379885958374688091, −5.13673343191472403190645370987, −3.25013411958641997846080051288, −2.22237919723525454571296753460, −1.40219991784540769690227806909, 1.486229296521182660425730720211, 2.3643750230742031439067178924, 3.56426224938052368867006258988, 5.123780905546805631578508619603, 6.738568211523039627253289222809, 7.34067873878773445073218512827, 8.37886539975332649891586465175, 9.47758557203473865080070022725, 10.05301496404914437774125212111, 10.88693356094751533676674676764, 12.18537672335958991174516047516, 13.77536346592574175897866253529, 14.25414668640404660036874389124, 15.11093133292522532752226535979, 16.447037566089139141502125345201, 16.99035385536220683869999633278, 18.342272338827547567451052024114, 18.64451714702623699981612560442, 19.90473653859569751954194904360, 20.59306748054923091898353000570, 21.21786718557113647446543771104, 22.32234345404203014497600816404, 23.83681271933273367741267522758, 24.64379447335672033374584371061, 25.41829670367451488721447803077

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