L-function 1-1368-1368.1307-r1-0-0

• Label: 1-1368-1368.1307-r1-0-0
• Degree: $1$
• Conductor: $1368$
• Sign: $-0.730 - 0.683i$
• 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

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.730 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign:	$-0.730 - 0.683i$
Analytic conductor:	\(147.012\)
Root analytic conductor:	\(147.012\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1368} (1307, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 1368,\ (1:\ ),\ -0.730 - 0.683i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.4509533235 - 1.141853814i\)
\(L(1)\)	\(\approx\)	\(1.087342787 - 0.1490475987i\)

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	\( 1 + (0.766 + 0.642i)T \)
	7	\( 1 + (0.5 - 0.866i)T \)
	11	\( 1 - T \)
	13	\( 1 + (0.766 - 0.642i)T \)
	17	\( 1 + (-0.766 - 0.642i)T \)
	23	\( 1 + (-0.939 - 0.342i)T \)
	29	\( 1 + (0.939 + 0.342i)T \)
	31	\( 1 + T \)
	37	\( 1 + T \)

Imaginary part of the first few zeros on the critical line

−21.17236397935551272520013854525, −20.33280181539954574160941266697, −19.47218043807652809539368127708, −18.42482467717539956308664607231, −18.019600244816433990252510002742, −17.30089619023247293489200272485, −16.304044624766469718970805984263, −15.707787553688748303393467878586, −14.943541601586156958596612648811, −13.860979124678011851940663531625, −13.391366611460304897044173266965, −12.53626026879873959564253967936, −11.76676170247356679811658481330, −10.90925925994850647321487605753, −10.00871520548034422509564437837, −9.220565107152399716071103243459, −8.38878866439827938602011496364, −7.94977181343526354621169033434, −6.32670791342327634777088410599, −6.021667118012428385390309536176, −4.92113251468197216321324924062, −4.37315584863164199406205972035, −2.893944359316743591777612931398, −2.052540720757795583761022902799, −1.29232722260136744489728607811, 0.218653910338166797125506432849, 1.37104970240759811609361730191, 2.43663112924168335226517464071, 3.20279336758973997447192373286, 4.374755555656500706409082642451, 5.19460791975796115901825007467, 6.1608266309944334947078913049, 6.87796201929661662901402460539, 7.82825373662223337523592973292, 8.47614235247348664815841402064, 9.707212067910739838190989287494, 10.40724591108114713910707046339, 10.8505991750535443899105889490, 11.74136058201760667126024027820, 12.99514477562428098099977596152, 13.55994739450182336397901524453, 14.08345038041889873898634416028, 15.015908116967024527337499464943, 15.81386445362534224452016508241, 16.57938325691106180158890529333, 17.74054160484285345739280018492, 17.9028518469353820071457982728, 18.61993572592999785387762964681, 19.78068320666714859407048418584, 20.45956845571375157950880800700

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