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Properties

Label 1-3895-3895.1013-r0-0-0
Degree $1$
Conductor $3895$
Sign $-0.260 + 0.965i$
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  + (0.882 + 0.469i)2-s + (0.996 − 0.0871i)3-s + (0.559 + 0.829i)4-s + (0.920 + 0.390i)6-s + (0.0523 + 0.998i)7-s + (0.104 + 0.994i)8-s + (0.984 − 0.173i)9-s + (0.629 − 0.777i)11-s + (0.629 + 0.777i)12-s + (−0.920 − 0.390i)13-s + (−0.422 + 0.906i)14-s + (−0.374 + 0.927i)16-s + (−0.325 + 0.945i)17-s + (0.951 + 0.309i)18-s + (0.139 + 0.990i)21-s + (0.920 − 0.390i)22-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)2-s + (0.996 − 0.0871i)3-s + (0.559 + 0.829i)4-s + (0.920 + 0.390i)6-s + (0.0523 + 0.998i)7-s + (0.104 + 0.994i)8-s + (0.984 − 0.173i)9-s + (0.629 − 0.777i)11-s + (0.629 + 0.777i)12-s + (−0.920 − 0.390i)13-s + (−0.422 + 0.906i)14-s + (−0.374 + 0.927i)16-s + (−0.325 + 0.945i)17-s + (0.951 + 0.309i)18-s + (0.139 + 0.990i)21-s + (0.920 − 0.390i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3895\)    =    \(5 \cdot 19 \cdot 41\)
Sign: $-0.260 + 0.965i$
Analytic conductor: \(18.0883\)
Root analytic conductor: \(18.0883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3895} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3895,\ (0:\ ),\ -0.260 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.828881600 + 3.693619254i\)
\(L(\frac12)\) \(\approx\) \(2.828881600 + 3.693619254i\)
\(L(1)\) \(\approx\) \(2.203737558 + 1.176274221i\)
\(L(1)\) \(\approx\) \(2.203737558 + 1.176274221i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.882 + 0.469i)T \)
3 \( 1 + (0.996 - 0.0871i)T \)
7 \( 1 + (0.0523 + 0.998i)T \)
11 \( 1 + (0.629 - 0.777i)T \)
13 \( 1 + (-0.920 - 0.390i)T \)
17 \( 1 + (-0.325 + 0.945i)T \)
23 \( 1 + (0.788 + 0.615i)T \)
29 \( 1 + (0.945 - 0.325i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
43 \( 1 + (0.0348 + 0.999i)T \)
47 \( 1 + (0.601 + 0.798i)T \)
53 \( 1 + (-0.981 - 0.190i)T \)
59 \( 1 + (0.882 + 0.469i)T \)
61 \( 1 + (-0.999 - 0.0348i)T \)
67 \( 1 + (0.945 - 0.325i)T \)
71 \( 1 + (-0.981 + 0.190i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.0871 - 0.996i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (0.224 + 0.974i)T \)
97 \( 1 + (0.999 + 0.0174i)T \)
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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

−18.61063548613437268112921464368, −17.7207874167065659718704269855, −16.80728156235331033288353843208, −16.12448970210050599926934637269, −15.331077097159691031144294379727, −14.69532592513821430920436915910, −14.04352447972198459520941898411, −13.8708991204595075695013258719, −12.78088604183849251183581616466, −12.438663428407319737942255917856, −11.532546475255333581423191035686, −10.652614320634832360495816083560, −10.10507681469183135377966377158, −9.402835202281598467577426993603, −8.79502989031210447838920066872, −7.43212736150117224173004609622, −7.12288122999548313047643770630, −6.56740383219420743083090910829, −5.0521847561420326669286103787, −4.68384381463828439436482484212, −3.936644849293648097253829909317, −3.280143932381177800398202643936, −2.377169344144298311040267817577, −1.78566500385530571241369933778, −0.7750131774257239363442594989, 1.416693684396338244402503713711, 2.28016354691296330700648087395, 2.98111033520976406606545634165, 3.54152281801780208766535670539, 4.466260168603644060988359761182, 5.17834767529327241224462580105, 6.07928927280562985984574576543, 6.63268627466857726429382356095, 7.59154586075674766329608419814, 8.14994016261974197053864472099, 8.84662508720975813810269109621, 9.40434585336698917361559129044, 10.48693680997702537131150488156, 11.40701645956781104838765378778, 12.08937963216464415841463184282, 12.78997833023173180553943679841, 13.2347451869044853641578170466, 14.171395359311876821504303832233, 14.58011341404048752677639127609, 15.26517886632088476515964342201, 15.62579161614779873829683697550, 16.51798256819533926030183577542, 17.30259883532125278273683235034, 17.93506353870740978298470163375, 19.06832705258542090627061975832

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