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Properties

Label 1-3751-3751.1264-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.0958 - 0.995i$
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  + (0.362 − 0.931i)2-s + (−0.913 + 0.406i)3-s + (−0.736 − 0.676i)4-s + (0.928 − 0.371i)5-s + (0.0475 + 0.998i)6-s + (0.999 − 0.0190i)7-s + (−0.897 + 0.441i)8-s + (0.669 − 0.743i)9-s + (−0.00951 − 0.999i)10-s + (0.948 + 0.318i)12-s + (0.861 + 0.508i)13-s + (0.345 − 0.938i)14-s + (−0.696 + 0.717i)15-s + (0.0855 + 0.996i)16-s + (0.879 + 0.475i)17-s + (−0.449 − 0.893i)18-s + ⋯
L(s)  = 1  + (0.362 − 0.931i)2-s + (−0.913 + 0.406i)3-s + (−0.736 − 0.676i)4-s + (0.928 − 0.371i)5-s + (0.0475 + 0.998i)6-s + (0.999 − 0.0190i)7-s + (−0.897 + 0.441i)8-s + (0.669 − 0.743i)9-s + (−0.00951 − 0.999i)10-s + (0.948 + 0.318i)12-s + (0.861 + 0.508i)13-s + (0.345 − 0.938i)14-s + (−0.696 + 0.717i)15-s + (0.0855 + 0.996i)16-s + (0.879 + 0.475i)17-s + (−0.449 − 0.893i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0958 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0958 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.0958 - 0.995i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1264, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.0958 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.581368780 - 1.436455384i\)
\(L(\frac12)\) \(\approx\) \(1.581368780 - 1.436455384i\)
\(L(1)\) \(\approx\) \(1.123734810 - 0.5935518539i\)
\(L(1)\) \(\approx\) \(1.123734810 - 0.5935518539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.362 - 0.931i)T \)
3 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (0.928 - 0.371i)T \)
7 \( 1 + (0.999 - 0.0190i)T \)
13 \( 1 + (0.861 + 0.508i)T \)
17 \( 1 + (0.879 + 0.475i)T \)
19 \( 1 + (-0.380 - 0.924i)T \)
23 \( 1 + (-0.516 + 0.856i)T \)
29 \( 1 + (0.610 - 0.791i)T \)
37 \( 1 + (0.995 + 0.0950i)T \)
41 \( 1 + (-0.988 + 0.151i)T \)
43 \( 1 + (-0.969 + 0.244i)T \)
47 \( 1 + (-0.362 - 0.931i)T \)
53 \( 1 + (-0.123 - 0.992i)T \)
59 \( 1 + (0.988 + 0.151i)T \)
61 \( 1 + (0.841 + 0.540i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (-0.432 + 0.901i)T \)
73 \( 1 + (0.797 + 0.603i)T \)
79 \( 1 + (0.345 - 0.938i)T \)
83 \( 1 + (-0.905 + 0.424i)T \)
89 \( 1 + (0.564 - 0.825i)T \)
97 \( 1 + (-0.985 - 0.170i)T \)
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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

−18.36824192561597432151159406848, −18.01897961986506060271384077485, −17.44123632733595071388932060732, −16.62378091109023722342904487325, −16.35039443317845404645059788990, −15.30181183153452082708999063064, −14.553842868108495981092110938037, −14.03745437032708353396986217935, −13.4138123588547778312231368438, −12.63060866695656074372868157953, −12.06924122284377343283339615771, −11.19236480159733834449426280843, −10.42486777109891577410457296958, −9.85158835825298047537922792484, −8.671850499643388794103255161594, −8.06725953923532906077082358643, −7.3914674882945982815009624003, −6.49266860384591327730860464806, −6.05606690065346064501685312338, −5.348054367148363191784794339565, −4.86000484862129438214295998665, −3.88676335060200248944413947542, −2.8328302115464118622565580606, −1.7363562707727604347745460300, −0.92728887535641554431999053238, 0.798448173889582526244791726872, 1.49903172547112745600104722790, 2.130408077372829009608124048639, 3.36851043419190178282897815923, 4.218021712686801175379011361159, 4.80647483993936638154134076886, 5.486236512177971350978729945925, 6.01832858008541670005363417724, 6.8183217986503821783637984834, 8.247397043227851242943302832452, 8.80413489053359442360270898393, 9.8074189263604690789138692314, 10.090283594449830798089562955588, 10.940909733211264437498814640991, 11.61252856290737717034864789108, 11.91062456190930575465654581183, 13.0491665732104811227558691693, 13.367927877484319617863933332814, 14.23869671577903333926394891055, 14.88975390351625272366936060403, 15.64449735640836722439959249890, 16.65515969234206561800902845756, 17.19169797833748198214488004409, 17.98196497648263864213201895599, 18.155516746164965433493414227962

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