L-function 1-3895-3895.3552-r1-0-0

• Label: 1-3895-3895.3552-r1-0-0
• Degree: $1$
• Conductor: $3895$
• Sign: $0.692 + 0.721i$
• Analytic cond.: $418.575$
• Root an. cond.: $418.575$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.309 − 0.951i)2^-s + (−0.707 − 0.707i)3^-s + (−0.809 − 0.587i)4^-s + (−0.891 + 0.453i)6^-s + (0.453 − 0.891i)7^-s + (−0.809 + 0.587i)8^-s + i·9^-s + (−0.156 + 0.987i)11^-s + (0.156 + 0.987i)12^-s + (−0.891 + 0.453i)13^-s + (−0.707 − 0.707i)14^-s + (0.309 + 0.951i)16^-s + (−0.156 + 0.987i)17^-s + (0.951 + 0.309i)18^-s + (−0.951 + 0.309i)21^-s + (0.891 + 0.453i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3895$    =    $5 \cdot 19 \cdot 41$
Sign:	$0.692 + 0.721i$
Analytic conductor:	$418.575$
Root analytic conductor:	$418.575$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3895} (3552, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3895,\ (1:\ ),\ 0.692 + 0.721i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3905171315 + 0.1664302600i$
$L(1)$	$\approx$	$0.6472952049 - 0.4906961803i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	19	$1$
	41	$1$
good	2	$1 + (0.309 - 0.951i)T$
	3	$1 + (-0.707 - 0.707i)T$
	7	$1 + (0.453 - 0.891i)T$
	11	$1 + (-0.156 + 0.987i)T$
	13	$1 + (-0.891 + 0.453i)T$
	17	$1 + (-0.156 + 0.987i)T$
	23	$1 + (0.951 + 0.309i)T$
	29	$1 + (0.987 - 0.156i)T$
	31	$1 + (-0.809 + 0.587i)T$

Imaginary part of the first few zeros on the critical line

−18.02468030183642806481067571806, −17.538013344321037865778474074219, −16.68365000110118335908319088162, −16.27920921451718890658437089294, −15.514370061181116229850902022463, −15.082780708334037779288295152595, −14.36909078530188074531231997940, −13.728157849549930292236675350658, −12.55483563431819286613510638335, −12.33633934377339923433904447459, −11.32593272457174419626624736454, −10.8470822821999418855759702394, −9.66779557874683193516602879285, −9.209977355603059556184935258406, −8.50723278656158634089104703017, −7.69959456667462547798277701129, −6.85793749329555352483249683933, −6.06276595413077961639380086191, −5.42211064653537690971843565142, −4.97749640025738446101492037357, −4.2991859859076184446662371000, −3.17674797911152665446522626840, −2.662582317947526411868426252980, −0.914335102871084953733639512670, −0.089342791099107408788256036507, 0.828767390620340625759562671922, 1.73327443933884126443901712426, 2.097325042742271975207961865683, 3.29074713599295268584924936296, 4.25040305021196835029956366451, 4.89987700336203148761205844937, 5.340669959676600431915389835948, 6.58834889726065701634743573067, 7.03057476758744639782975261369, 7.91668254896142824987919419198, 8.70176613836313092647297018124, 9.79935919767709640905166984213, 10.33842462043500764906067638491, 10.91247217652086987236226074470, 11.64508781533430957148704532428, 12.264801479014618524050514412880, 12.83460796292488472890707083001, 13.43288811069780240356128899748, 14.20089386716412680172696321094, 14.75579955475983644104020640987, 15.62237034942869674802478010464, 16.87356861268119247689641410643, 17.25371008926744418476568227827, 17.761799871645624107776343539645, 18.48372143901391334845139297256

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