L-function 1-95-95.52-r0-0-0

• Label: 1-95-95.52-r0-0-0
• Degree: $1$
• Conductor: $95$
• Sign: $-0.947 - 0.319i$
• 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

+ (−0.642 − 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

$\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$95$    =    $5 \cdot 19$
Sign:	$-0.947 - 0.319i$
Analytic conductor:	$0.441178$
Root analytic conductor:	$0.441178$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{95} (52, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 95,\ (0:\ ),\ -0.947 - 0.319i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.09844109826 - 0.6006086724i$
$L(1)$	$\approx$	$0.4828072977 - 0.5117430602i$

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

−31.12242189745336943663881945502, −29.111797334717695665069041376637, −28.3397504672060549765001326549, −27.43492636499187476278965415517, −26.2833459141826407300997222680, −25.75161252748203616811999877242, −24.78561449980639516465976326854, −23.28779892816688334085687053487, −22.42114462516063634819227172042, −21.12088404150241118535056115814, −19.86366893024309898905664800614, −18.98932745863336673835784642166, −17.70555875592805975237321806781, −16.37566181825622175381299470803, −15.82582928376133605319055899653, −14.7681842321104730504097958515, −13.71228689975848059076453384590, −11.87321659399841998948081112009, −10.13329951743833925893358381172, −9.64085565919297399579854665536, −8.48294765100832985696601358312, −7.095857806544854597440239975127, −5.65475188308628297460511608462, −4.42327703320778157428014223204, −2.49583477297755389244240077143, 0.74511025449881322592963152586, 2.57173992739758702876811472830, 3.61643546467546415615303384655, 6.05542231988309087738032703557, 7.539813154119690778266476623307, 8.35710282936524052313292993286, 9.7776332765217585382387141098, 10.87559206583373394574283181581, 12.3739037003792309603730347423, 13.01371531013627027091853761304, 14.12018602200271677687027188532, 16.01859806337734718550434024691, 17.1926162308816504811908489994, 18.20593215158568090874805481583, 19.25710292121462792534153560144, 19.82030796076300675170453470060, 20.9608113821875289654274933316, 22.3030005662048703505648376989, 23.39816853216645008033222697353, 24.69857658447398320121344084967, 25.86640245598931892059791984344, 26.42763306348777751096200622179, 27.80531827893020219389296716582, 29.000402725331087728262504296251, 29.674012696938874486313948172117

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