L-function 1-1368-1368.1051-r1-0-0

• Label: 1-1368-1368.1051-r1-0-0
• Degree: $1$
• Conductor: $1368$
• Sign: $-0.486 + 0.873i$
• Analytic cond.: $147.012$
• Root an. cond.: $147.012$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.173 + 0.984i)5^-s + (0.5 − 0.866i)7^-s + 11^-s + (−0.173 − 0.984i)13^-s + (0.173 − 0.984i)17^-s + (−0.766 + 0.642i)23^-s + (−0.939 − 0.342i)25^-s + (−0.766 + 0.642i)29^-s − 31^-s + (0.766 + 0.642i)35^-s − 37^-s + (−0.939 + 0.342i)41^-s + (0.766 + 0.642i)43^-s + (−0.766 + 0.642i)47^-s + (−0.5 − 0.866i)49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign:	$-0.486 + 0.873i$
Analytic conductor:	\(147.012\)
Root analytic conductor:	\(147.012\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1368} (1051, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 1368,\ (1:\ ),\ -0.486 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.5235318491 + 0.8911229036i\)
\(L(1)\)	\(\approx\)	\(0.9868389694 + 0.08724457152i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	3	\( 1 \)
	19	\( 1 \)
good	5	\( 1 + (-0.173 + 0.984i)T \)
	7	\( 1 + (0.5 - 0.866i)T \)
	11	\( 1 + T \)
	13	\( 1 + (-0.173 - 0.984i)T \)
	17	\( 1 + (0.173 - 0.984i)T \)
	23	\( 1 + (-0.766 + 0.642i)T \)
	29	\( 1 + (-0.766 + 0.642i)T \)
	31	\( 1 - T \)
	37	\( 1 - T \)

Imaginary part of the first few zeros on the critical line

−20.50308318392921289775273372740, −19.574459632566247343195471287895, −19.05724290450882183794727871649, −18.14800900312667608848504649564, −17.20023663932468962228012256246, −16.7285433818484182046932163404, −15.93515038569235788407842494898, −15.00191959804481511916458279702, −14.43358292988494516114024921267, −13.50026312995658923458095221200, −12.5263944093992844318619052442, −11.95808573909420421127672985439, −11.448143665255777015037915167152, −10.245906801471180359599249395277, −9.215402013232015221443602999211, −8.77867162769110355963900489367, −8.06555507722494575344995550032, −6.96415787111753138196342539227, −5.98095076873913446228964775716, −5.271509847379219981133960080833, −4.27585842052320812535471036092, −3.69550179670141572672332370769, −2.004922650469057946295619364067, −1.6513726951213701792342637110, −0.20792557574403803609802921419, 1.00582706290702926131632549542, 2.066466422515955842543300974478, 3.307642275866502572823900026218, 3.78681339743359192036149054381, 4.928749358187692254695500294517, 5.86900336442528123276271368447, 6.93858789668095763691414826343, 7.395000027583960587760836534803, 8.1978280112571808606648935389, 9.39975664930105052591030883027, 10.10987680027161046370527577996, 10.94657584156424631216839494326, 11.478698438875376503482632370976, 12.35919901780270154181341431984, 13.466038281334348599023995268965, 14.16691452488970246779790679627, 14.67242512030181294310366522437, 15.48798550177205118584530005341, 16.41879840276482917971249090811, 17.22909532009477253955488047301, 17.95538391534285958282156666202, 18.480391273115669597366515625400, 19.6560273516193062675636080219, 19.95828828724128205488485088667, 20.84319876851340424852745127226

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