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Properties

Label 1-3751-3751.2124-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.249 + 0.968i$
Analytic cond. $17.4195$
Root an. cond. $17.4195$
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  + (−0.254 − 0.967i)2-s + (0.309 + 0.951i)3-s + (−0.870 + 0.491i)4-s + (−0.142 + 0.989i)5-s + (0.841 − 0.540i)6-s + (0.974 − 0.226i)7-s + (0.696 + 0.717i)8-s + (−0.809 + 0.587i)9-s + (0.993 − 0.113i)10-s + (−0.736 − 0.676i)12-s + (0.993 + 0.113i)13-s + (−0.466 − 0.884i)14-s + (−0.985 + 0.170i)15-s + (0.516 − 0.856i)16-s + (0.941 − 0.336i)17-s + (0.774 + 0.633i)18-s + ⋯
L(s)  = 1  + (−0.254 − 0.967i)2-s + (0.309 + 0.951i)3-s + (−0.870 + 0.491i)4-s + (−0.142 + 0.989i)5-s + (0.841 − 0.540i)6-s + (0.974 − 0.226i)7-s + (0.696 + 0.717i)8-s + (−0.809 + 0.587i)9-s + (0.993 − 0.113i)10-s + (−0.736 − 0.676i)12-s + (0.993 + 0.113i)13-s + (−0.466 − 0.884i)14-s + (−0.985 + 0.170i)15-s + (0.516 − 0.856i)16-s + (0.941 − 0.336i)17-s + (0.774 + 0.633i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.249 + 0.968i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (2124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.249 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.449322208 + 1.122726320i\)
\(L(\frac12)\) \(\approx\) \(1.449322208 + 1.122726320i\)
\(L(1)\) \(\approx\) \(1.111250094 + 0.2066420370i\)
\(L(1)\) \(\approx\) \(1.111250094 + 0.2066420370i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.254 - 0.967i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (0.974 - 0.226i)T \)
13 \( 1 + (0.993 + 0.113i)T \)
17 \( 1 + (0.941 - 0.336i)T \)
19 \( 1 + (-0.0285 + 0.999i)T \)
23 \( 1 + (0.974 + 0.226i)T \)
29 \( 1 + (-0.0285 + 0.999i)T \)
37 \( 1 + (0.415 + 0.909i)T \)
41 \( 1 + (-0.254 - 0.967i)T \)
43 \( 1 + (-0.985 - 0.170i)T \)
47 \( 1 + (-0.254 + 0.967i)T \)
53 \( 1 + (0.0855 - 0.996i)T \)
59 \( 1 + (-0.254 + 0.967i)T \)
61 \( 1 + (0.841 + 0.540i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.610 - 0.791i)T \)
73 \( 1 + (0.0855 + 0.996i)T \)
79 \( 1 + (-0.466 - 0.884i)T \)
83 \( 1 + (0.516 + 0.856i)T \)
89 \( 1 + (0.610 + 0.791i)T \)
97 \( 1 + (-0.466 + 0.884i)T \)
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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

−18.494792310146544163058472163213, −17.6103618371824707752199493041, −17.22857692081556452259295720817, −16.56685096526239773859299672184, −15.68238259005447807346656493923, −15.0391559903767491138393808882, −14.441871193770310667113329594608, −13.59428422789664696949076751174, −13.17636956248810672157657915609, −12.50779355536992208467187351816, −11.60094769828798535777669561407, −10.98187792444282860677697575079, −9.73671716525758476808732303971, −9.0164834413482799346150729277, −8.36265921708589011779955310334, −8.096426396001105547340731144824, −7.32322449279631012834179744884, −6.483424328042968108966270569796, −5.69438260211949866946447778568, −5.139566638687312989262603668271, −4.31432656092913689143517267277, −3.372464212383644803066873563819, −2.055513183342233606588212123266, −1.21915951410298996948308814867, −0.67062822321977011565279904955, 1.13992816360260435397020371933, 1.965134195791157738362904565558, 2.97316469018568640588766006681, 3.50281662068618268357777927287, 4.07268768072572643988039462970, 5.02562755044722279249737397263, 5.621744876590704564264985050250, 6.90711781923438411557672541515, 7.89758621154929480095702402215, 8.28249521790498326944089204731, 9.11776939667124119777425128790, 9.94706213677655161519055121574, 10.47353574127736171990839463288, 11.06327910232970167907720708985, 11.494767338735399972367971540102, 12.26961155677378818762864205184, 13.47574817540503919653718592021, 13.96450744844189529481059491421, 14.62439035450968253345485812100, 15.05079876356506771226229665176, 16.147205776340754736201787113985, 16.753179410585694088076525970447, 17.51625396831636333646692617501, 18.29047424399817144189799991051, 18.76233752381725157720866942194

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