L-function 1-95-95.27-r0-0-0

• Label: 1-95-95.27-r0-0-0
• Degree: $1$
• Conductor: $95$
• Sign: $0.991 + 0.127i$
• Analytic cond.: $0.441178$
• Root an. cond.: $0.441178$
• 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 + (−0.5 + 0.866i)6^-s + i·7^-s + i·8^-s + (0.5 − 0.866i)9^-s + 11^-s − i·12^-s + (−0.866 − 0.5i)13^-s + (−0.5 − 0.866i)14^-s + (−0.5 − 0.866i)16^-s + (0.866 − 0.5i)17^-s + i·18^-s + (0.5 + 0.866i)21^-s + (−0.866 + 0.5i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$95$    =    $5 \cdot 19$
Sign:	$0.991 + 0.127i$
Analytic conductor:	$0.441178$
Root analytic conductor:	$0.441178$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{95} (27, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 95,\ (0:\ ),\ 0.991 + 0.127i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.9191528630 + 0.05884527032i$
$L(1)$	$\approx$	$0.9452109554 + 0.06355692387i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−30.14478302374215218367829815622, −29.19933423769492854208036616345, −27.76103082598314813172572564066, −27.05535020422464662250435147172, −26.27184439362836933090795477601, −25.32732382418260580458770294862, −24.25721085818293273949101388628, −22.45812984064728908507401614377, −21.38171313516966552259827427514, −20.44026407351372806135985740228, −19.60280399375680085570424207946, −18.85181449382051277910892743847, −17.06079801642375459489721714729, −16.63811213004985408389921665450, −15.03658561919824872152800397615, −13.95123142725517641567263700722, −12.578807459511616275393282226606, −11.13206117608775653126856791853, −10.01392754016456614584385853345, −9.21934367625891441188778725846, −7.93085360561015145759414073561, −6.925612721765482170710184870257, −4.34338036253021673064181784620, −3.280817955322843498243209889044, −1.64524898387273256848166656262, 1.58348682469929495762951840284, 3.01352895224316356312594537840, 5.38199802318896379831780062290, 6.80383117295982513791911911156, 7.83223289911116261247481629873, 9.02653878264752100285167661727, 9.67845695439024174604692812303, 11.52789204258280998488223151785, 12.713539649326962386350214632900, 14.49111739253738591783252154434, 14.89008610455928670641290497733, 16.25958193128096865105953996142, 17.58513925508679577952902131993, 18.55868239202358882996257963073, 19.40359210175515103409499878287, 20.250114422530522082530394775336, 21.62961250893157715305055937495, 23.191186253814226835790871574621, 24.607917414055833160359005743663, 24.96570242834065587295872955955, 25.85314589376245965589644624706, 27.11678231071668760008881723454, 27.808525068609996749905408080657, 29.239006686510723176550355395463, 29.999784790118791622389068102405

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