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Properties

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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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

\(L(\frac{1}{2})\) \(\approx\) \(0.3905171315 + 0.1664302600i\)
\(L(\frac12)\) \(\approx\) \(0.3905171315 + 0.1664302600i\)
\(L(1)\) \(\approx\) \(0.6472952049 - 0.4906961803i\)
\(L(1)\) \(\approx\) \(0.6472952049 - 0.4906961803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
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 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (-0.453 - 0.891i)T \)
53 \( 1 + (-0.156 - 0.987i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (0.987 - 0.156i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.891 - 0.453i)T \)
97 \( 1 + (-0.987 + 0.156i)T \)
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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

−18.02468030183642806481067571806, −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