L-function 1-3895-3895.1079-r0-0-0

• Label: 1-3895-3895.1079-r0-0-0
• Degree: $1$
• Conductor: $3895$
• Sign: $-0.205 + 0.978i$
• Analytic cond.: $18.0883$
• Root an. cond.: $18.0883$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.0697 + 0.997i)2^-s + (0.906 + 0.422i)3^-s + (−0.990 − 0.139i)4^-s + (−0.484 + 0.874i)6^-s + (−0.777 + 0.629i)7^-s + (0.207 − 0.978i)8^-s + (0.642 + 0.766i)9^-s + (−0.544 − 0.838i)11^-s + (−0.838 − 0.544i)12^-s + (−0.874 − 0.484i)13^-s + (−0.573 − 0.819i)14^-s + (0.961 + 0.275i)16^-s + (0.390 + 0.920i)17^-s + (−0.809 + 0.587i)18^-s + (−0.970 + 0.241i)21^-s + (0.874 − 0.484i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3895$    =    $5 \cdot 19 \cdot 41$
Sign:	$-0.205 + 0.978i$
Analytic conductor:	$18.0883$
Root analytic conductor:	$18.0883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3895} (1079, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3895,\ (0:\ ),\ -0.205 + 0.978i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.9957637965 + 1.226308930i$
$L(1)$	$\approx$	$0.8841903630 + 0.6308122644i$

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	$1 + (-0.0697 + 0.997i)T$
	3	$1 + (0.906 + 0.422i)T$
	7	$1 + (-0.777 + 0.629i)T$
	11	$1 + (-0.544 - 0.838i)T$
	13	$1 + (-0.874 - 0.484i)T$
	17	$1 + (0.390 + 0.920i)T$
	23	$1 + (-0.719 - 0.694i)T$
	29	$1 + (0.390 - 0.920i)T$
	31	$1 + (0.669 - 0.743i)T$

Imaginary part of the first few zeros on the critical line

−18.48427706466980854789970994056, −17.96525309719182200468202080209, −17.19095794684065170735568738296, −16.36443625310158260893089216550, −15.50413926830434648640309638173, −14.668840492288799479370609138572, −13.97909839218374460182371378358, −13.517181024992393270619842092177, −12.85736305728438468378696235806, −12.141576569188517521654922798901, −11.762291969343161034068712077760, −10.42249671831855543374210358932, −9.95829461345470069937536128497, −9.54432822045062755251443056343, −8.72940954748683329159126662119, −7.878500188049161560471801769246, −7.2322297354162650695726878884, −6.64150434164101769655668988350, −5.232410251590948756614569931047, −4.579395984283221605907197096532, −3.69669450973158595094113243905, −3.07348070857568819031859567749, −2.37467913005052109832140659628, −1.63232013252839764893108432008, −0.62157418713710440707494803797, 0.67116366721676320829069972465, 2.23267461303179010934279706231, 2.89964648698624986506758470167, 3.751425590520581268615692478271, 4.44429326104795569260189854026, 5.41623487893872265689209045110, 5.96348226096952274830515621419, 6.74939199225450621456063764959, 7.871839881174970389579017200489, 8.07488687256134113308571194189, 8.83425514793606665979501875792, 9.65666998115671741376526263322, 10.0356773246007466660298303741, 10.802751302352988982089378042263, 12.202985684042071969152901586079, 12.77511401163576602153114477389, 13.454099416319510991334304322243, 14.06576970125320546233530875011, 14.858092025408436600786566537209, 15.29637341819808252423166164042, 15.91340239694681700120539489531, 16.50256445381535255610098917170, 17.12899052225762732124045839302, 18.12678406521377209972919918842, 18.76238348633286920706346098559

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