L-function 1-95-95.23-r1-0-0

• Label: 1-95-95.23-r1-0-0
• Degree: $1$
• Conductor: $95$
• Sign: $-0.439 + 0.898i$
• 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

+ (0.642 − 0.766i)2^-s + (−0.342 − 0.939i)3^-s + (−0.173 − 0.984i)4^-s + (−0.939 − 0.342i)6^-s + (−0.866 + 0.5i)7^-s + (−0.866 − 0.5i)8^-s + (−0.766 + 0.642i)9^-s + (−0.5 + 0.866i)11^-s + (−0.866 + 0.5i)12^-s + (0.342 − 0.939i)13^-s + (−0.173 + 0.984i)14^-s + (−0.939 + 0.342i)16^-s + (0.642 − 0.766i)17^-s + i·18^-s + (0.766 + 0.642i)21^-s + (0.342 + 0.939i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$95$    =    $5 \cdot 19$
Sign:	$-0.439 + 0.898i$
Analytic conductor:	$10.2091$
Root analytic conductor:	$10.2091$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{95} (23, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 95,\ (1:\ ),\ -0.439 + 0.898i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.3497137382 - 0.5601976959i$
$L(1)$	$\approx$	$0.5965687012 - 0.6674330367i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	19	$1$
good	2	$1 + (0.642 - 0.766i)T$
	3	$1 + (-0.342 - 0.939i)T$
	7	$1 + (-0.866 + 0.5i)T$
	11	$1 + (-0.5 + 0.866i)T$
	13	$1 + (0.342 - 0.939i)T$
	17	$1 + (0.642 - 0.766i)T$
	23	$1 + (-0.984 + 0.173i)T$
	29	$1 + (-0.766 + 0.642i)T$
	31	$1 + (-0.5 - 0.866i)T$

Imaginary part of the first few zeros on the critical line

−30.94243604674977948958265943358, −29.59961267039313672724218129057, −28.612088658561208255227119446584, −27.21763913980803072457037691183, −26.20557605588313276956830417079, −25.79761619061461641226604458162, −24.024826142751495068540501580748, −23.34630296804755204400924939421, −22.24801989763059694770963550749, −21.49014426037987623326318389887, −20.47830002353351941271124702323, −18.861104045681340629236942607263, −17.25712284835781084240895170859, −16.38373637406579384791446979359, −15.82055064072091974407518211577, −14.476300880006309499456367359349, −13.49174911312073558430473760731, −12.15208694490535158407410267678, −10.83807026633224549410073409505, −9.494898084952516155217991840307, −8.206187441113791849551139205205, −6.53891879084937925916689457559, −5.640452464303071769200658192996, −4.16116066900008687923313443228, −3.269918589202958451741776769457, 0.23710456500330123924622987430, 2.01429348739387199417122852525, 3.25264545885616451345748397835, 5.22567953447397373557343620678, 6.14767269563634193962204681315, 7.59744136371661406243118357876, 9.40843568354654126379013416789, 10.62311526621822162643543396916, 11.97666966626212887907224754885, 12.7059488447199990192992140839, 13.520406887732427875352736950902, 14.90959489505560074119572099520, 16.16937018266751020037620948853, 17.95988554663793216990438418929, 18.59213269299840669682264725015, 19.74042597768436183267266145414, 20.60286308463406481440791743138, 22.18260642291360935154385210523, 22.82286385365912383263584272293, 23.68555806739592763704933859575, 24.91526869847997030433189077588, 25.776097562100778979021556828322, 27.79362144393399565653195309370, 28.39287880792974354802535081038, 29.46334248797391558856752942881

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