L-function 1-95-95.69-r1-0-0

• Label: 1-95-95.69-r1-0-0
• Degree: $1$
• Conductor: $95$
• Sign: $0.910 + 0.412i$
• Analytic cond.: $10.2091$
• Root an. cond.: $10.2091$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.5 + 0.866i)2^-s + (−0.5 + 0.866i)3^-s + (−0.5 − 0.866i)4^-s + (−0.5 − 0.866i)6^-s − 7^-s + 8^-s + (−0.5 − 0.866i)9^-s + 11^-s + 12^-s + (−0.5 − 0.866i)13^-s + (0.5 − 0.866i)14^-s + (−0.5 + 0.866i)16^-s + (0.5 − 0.866i)17^-s + 18^-s + (0.5 − 0.866i)21^-s + (−0.5 + 0.866i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$95$    =    $5 \cdot 19$
Sign:	$0.910 + 0.412i$
Analytic conductor:	$10.2091$
Root analytic conductor:	$10.2091$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{95} (69, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 95,\ (1:\ ),\ 0.910 + 0.412i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7493325977 + 0.1619510970i$
$L(1)$	$\approx$	$0.5871051878 + 0.2662138030i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−29.74966261505731444281302099449, −28.83894445812640638535875266831, −28.20626795668210430280522992642, −26.89997357630257977515182690369, −25.76949348832090165346543984798, −24.78590510717837387987223039755, −23.33496306201211115476656486483, −22.41049614845421594339612838109, −21.53831678171901603493006431512, −19.83373899034294888240598593538, −19.286058874785705998691599616115, −18.3887552136322795391384216772, −17.026946294358975562711379925390, −16.55723773771032536640535227131, −14.32218177708608255235017633188, −13.03576887056176117008546252422, −12.27556202285972215033199254752, −11.29525380683775317781212120902, −9.957405876840713118717771544893, −8.79022316154758060202808409492, −7.32504309729048522299432997256, −6.20171853055659209316800961221, −4.18930032711182666460589212286, −2.550530449512920603019314588028, −1.060958596821136263492070983440, 0.55805807764059027320931887862, 3.48658840108925729744621219309, 5.05189825357740803548480255399, 6.11573377234112734838858333664, 7.30923577058409387829414276624, 9.118433287416195014042408156910, 9.711674132828103668734264782389, 10.91055972367194092521789613937, 12.47033963688025007383484740907, 14.11580275800595363554161141214, 15.196946149288540004136961562881, 16.13494534404999995668951490903, 16.92688130404687464087442286197, 17.90256257262080779985128565353, 19.31351249207235006511626653061, 20.27872269258700822284501091280, 22.0515037880881388322529285260, 22.640610059556049767791609854250, 23.63422119695295554647229646289, 25.13997025493791388024563756041, 25.758261657386634641099952199355, 27.22069767797567502113412896233, 27.42091028249805495963153496881, 28.778452762143196522714278020939, 29.58405930363759882008322444684

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