L-function 1-385-385.314-r0-0-0

• Label: 1-385-385.314-r0-0-0
• Degree: $1$
• Conductor: $385$
• Sign: $-0.605 + 0.795i$
• Analytic cond.: $1.78793$
• Root an. cond.: $1.78793$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.809 + 0.587i)2^-s + (0.309 + 0.951i)3^-s + (0.309 − 0.951i)4^-s + (−0.809 − 0.587i)6^-s + (0.309 + 0.951i)8^-s + (−0.809 + 0.587i)9^-s + 12^-s + (0.809 − 0.587i)13^-s + (−0.809 − 0.587i)16^-s + (0.809 + 0.587i)17^-s + (0.309 − 0.951i)18^-s + (0.309 + 0.951i)19^-s − 23^-s + (−0.809 + 0.587i)24^-s + (−0.309 + 0.951i)26^-s + (−0.809 − 0.587i)27^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign:	$-0.605 + 0.795i$
Analytic conductor:	\(1.78793\)
Root analytic conductor:	\(1.78793\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{385} (314, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 385,\ (0:\ ),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.4083598073 + 0.8237125192i\)
\(L(1)\)	\(\approx\)	\(0.6599041163 + 0.4850398364i\)

Euler product

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

	$p$	$F_p(T)$
bad	5	\( 1 \)
	7	\( 1 \)
	11	\( 1 \)
good	2	\( 1 + (-0.809 + 0.587i)T \)
	3	\( 1 + (0.309 + 0.951i)T \)
	13	\( 1 + (0.809 - 0.587i)T \)
	17	\( 1 + (0.809 + 0.587i)T \)
	19	\( 1 + (0.309 + 0.951i)T \)
	23	\( 1 - T \)
	29	\( 1 + (-0.309 + 0.951i)T \)
	31	\( 1 + (0.809 - 0.587i)T \)
	37	\( 1 + (-0.309 + 0.951i)T \)

Imaginary part of the first few zeros on the critical line

−24.41937131415375673094527930689, −23.4030359750567613000005659735, −22.4393192157669017329788577759, −21.22197277243873843329453577445, −20.56964033858555178169359150171, −19.612163794578381885233307798843, −18.98387841784789964083391808397, −18.16082878057215838248427234903, −17.53546788989464440152798407926, −16.46737602613856668734791792515, −15.53635709099357449358022840816, −14.01544110284492381555363417121, −13.45117560776970288781810090302, −12.25150048466594068622158915872, −11.727093691421027092851235281291, −10.68571068056962627249812335744, −9.44190534862250200563407040906, −8.720252637004305841114266632210, −7.73146068561553044325461203729, −6.984441014902998757750248198686, −5.86765235189618134099112644476, −4.04039954075135206774990722128, −2.91218792223786168098219154995, −1.922888085920635106107230175618, −0.75178584337804524846713209451, 1.42204292788780681480340909884, 2.97704034293110944088297969613, 4.184181486530333075540767667215, 5.503581410707410954899337535097, 6.15139680602169060792624339704, 7.78097683795823430324897863524, 8.298573046549319451796722072959, 9.37004411149672321099563850217, 10.201615387360807480798028978806, 10.8182108184482719944016872455, 12.0115812363506588253110204394, 13.59798617652472531636567250923, 14.48541066814588587607501329516, 15.21388423142382840919830109891, 16.11498432795115069046388233628, 16.660217537824930462962943544935, 17.694916648877549816054612646200, 18.623024563522490798283999087779, 19.54787275460025494163979469418, 20.45395039574418069878908400480, 21.03909878344550352917136205669, 22.33257999907677308949202195787, 23.09467695387331780173742995196, 24.09908241089036441453300194497, 25.15493057302727861866549974420

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