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Properties

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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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

\(L(\frac{1}{2})\) \(\approx\) \(0.9957637965 + 1.226308930i\)
\(L(\frac12)\) \(\approx\) \(0.9957637965 + 1.226308930i\)
\(L(1)\) \(\approx\) \(0.8841903630 + 0.6308122644i\)
\(L(1)\) \(\approx\) \(0.8841903630 + 0.6308122644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.0697 + 0.997i)T \)
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 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.898 + 0.438i)T \)
47 \( 1 + (0.515 + 0.857i)T \)
53 \( 1 + (0.601 + 0.798i)T \)
59 \( 1 + (0.997 + 0.0697i)T \)
61 \( 1 + (-0.898 - 0.438i)T \)
67 \( 1 + (0.920 + 0.390i)T \)
71 \( 1 + (0.798 + 0.601i)T \)
73 \( 1 + (-0.342 - 0.939i)T \)
79 \( 1 + (0.906 + 0.422i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.190 + 0.981i)T \)
97 \( 1 + (0.224 - 0.974i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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