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Properties

Label 1-3895-3895.1088-r0-0-0
Degree $1$
Conductor $3895$
Sign $0.0295 - 0.999i$
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  + (−0.997 + 0.0697i)2-s + (−0.422 − 0.906i)3-s + (0.990 − 0.139i)4-s + (0.484 + 0.874i)6-s + (−0.629 + 0.777i)7-s + (−0.978 + 0.207i)8-s + (−0.642 + 0.766i)9-s + (0.544 − 0.838i)11-s + (−0.544 − 0.838i)12-s + (0.484 + 0.874i)13-s + (0.573 − 0.819i)14-s + (0.961 − 0.275i)16-s + (−0.920 − 0.390i)17-s + (0.587 − 0.809i)18-s + (0.970 + 0.241i)21-s + (−0.484 + 0.874i)22-s + ⋯
L(s)  = 1  + (−0.997 + 0.0697i)2-s + (−0.422 − 0.906i)3-s + (0.990 − 0.139i)4-s + (0.484 + 0.874i)6-s + (−0.629 + 0.777i)7-s + (−0.978 + 0.207i)8-s + (−0.642 + 0.766i)9-s + (0.544 − 0.838i)11-s + (−0.544 − 0.838i)12-s + (0.484 + 0.874i)13-s + (0.573 − 0.819i)14-s + (0.961 − 0.275i)16-s + (−0.920 − 0.390i)17-s + (0.587 − 0.809i)18-s + (0.970 + 0.241i)21-s + (−0.484 + 0.874i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0295 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0295 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3895\)    =    \(5 \cdot 19 \cdot 41\)
Sign: $0.0295 - 0.999i$
Analytic conductor: \(18.0883\)
Root analytic conductor: \(18.0883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3895} (1088, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3895,\ (0:\ ),\ 0.0295 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3922559992 - 0.4040186015i\)
\(L(\frac12)\) \(\approx\) \(0.3922559992 - 0.4040186015i\)
\(L(1)\) \(\approx\) \(0.5283162255 - 0.1171934443i\)
\(L(1)\) \(\approx\) \(0.5283162255 - 0.1171934443i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.997 + 0.0697i)T \)
3 \( 1 + (-0.422 - 0.906i)T \)
7 \( 1 + (-0.629 + 0.777i)T \)
11 \( 1 + (0.544 - 0.838i)T \)
13 \( 1 + (0.484 + 0.874i)T \)
17 \( 1 + (-0.920 - 0.390i)T \)
23 \( 1 + (0.694 + 0.719i)T \)
29 \( 1 + (-0.390 - 0.920i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + (-0.438 + 0.898i)T \)
47 \( 1 + (-0.857 - 0.515i)T \)
53 \( 1 + (-0.798 - 0.601i)T \)
59 \( 1 + (0.997 - 0.0697i)T \)
61 \( 1 + (-0.898 + 0.438i)T \)
67 \( 1 + (0.390 + 0.920i)T \)
71 \( 1 + (0.798 - 0.601i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (-0.906 + 0.422i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.190 + 0.981i)T \)
97 \( 1 + (-0.974 + 0.224i)T \)
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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

−18.497151902034643894437148437673, −17.92543866567878480648538697198, −17.20161809009387977856965276767, −16.83319008837735313425070557776, −16.124472518497094958823304814806, −15.47949516410185311669240341322, −14.95486622630169521123069121688, −14.16423270046438473556129496547, −12.79936877270791894954747821708, −12.61511401392705956190560090902, −11.42750047453667844924033146593, −10.885851367327879592696739436480, −10.39330860463580657905664484024, −9.74601372090925321933675053075, −9.09527197669564938491006629063, −8.49548954723126650875628249872, −7.48055232066963192261615078957, −6.68430679744148168872324620474, −6.289725203550455751547669082300, −5.22800401308809884420657846584, −4.329535938378941395451980932431, −3.52124073330916276712912679952, −2.91643048042105872668148803738, −1.68270555696624042882124328425, −0.72203510226805053767138014208, 0.34469303463842568309413387603, 1.35672962643315084477091995270, 2.13392213599387674051943142566, 2.8304715578076783191002896470, 3.80456603168184459109214795848, 5.19964191137609530354533577916, 6.01767394457269885446816321627, 6.42880817777283697965007277750, 7.02915461771087238044882261899, 7.91812451897001988883409344566, 8.6374415038500405852585243960, 9.212208297401422769629592015605, 9.79187835581750263974954857393, 11.16827666005870001033247562848, 11.31313281600151806153516662271, 11.84421020477733845566931448712, 12.90321271086852602884753357407, 13.33625912466956765068389378138, 14.29887918483244146448100407409, 15.1192926084715378616850868481, 15.94971535178388862575658584518, 16.49010246298044414270622681209, 17.0139304017644603060082153388, 17.820159857166059307129641864444, 18.48975505992554872886257410794

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