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Properties

Label 1-3751-3751.2333-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.940 - 0.339i$
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  + (0.941 + 0.336i)2-s + (−0.809 + 0.587i)3-s + (0.774 + 0.633i)4-s + (−0.654 − 0.755i)5-s + (−0.959 + 0.281i)6-s + (−0.736 + 0.676i)7-s + (0.516 + 0.856i)8-s + (0.309 − 0.951i)9-s + (−0.362 − 0.931i)10-s + (−0.998 − 0.0570i)12-s + (−0.362 + 0.931i)13-s + (−0.921 + 0.389i)14-s + (0.974 + 0.226i)15-s + (0.198 + 0.980i)16-s + (0.897 + 0.441i)17-s + (0.610 − 0.791i)18-s + ⋯
L(s)  = 1  + (0.941 + 0.336i)2-s + (−0.809 + 0.587i)3-s + (0.774 + 0.633i)4-s + (−0.654 − 0.755i)5-s + (−0.959 + 0.281i)6-s + (−0.736 + 0.676i)7-s + (0.516 + 0.856i)8-s + (0.309 − 0.951i)9-s + (−0.362 − 0.931i)10-s + (−0.998 − 0.0570i)12-s + (−0.362 + 0.931i)13-s + (−0.921 + 0.389i)14-s + (0.974 + 0.226i)15-s + (0.198 + 0.980i)16-s + (0.897 + 0.441i)17-s + (0.610 − 0.791i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.940 - 0.339i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (2333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.940 - 0.339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1949638966 + 1.113389000i\)
\(L(\frac12)\) \(\approx\) \(-0.1949638966 + 1.113389000i\)
\(L(1)\) \(\approx\) \(0.9084076576 + 0.6095151046i\)
\(L(1)\) \(\approx\) \(0.9084076576 + 0.6095151046i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.941 + 0.336i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.654 - 0.755i)T \)
7 \( 1 + (-0.736 + 0.676i)T \)
13 \( 1 + (-0.362 + 0.931i)T \)
17 \( 1 + (0.897 + 0.441i)T \)
19 \( 1 + (-0.466 + 0.884i)T \)
23 \( 1 + (-0.736 - 0.676i)T \)
29 \( 1 + (-0.466 + 0.884i)T \)
37 \( 1 + (0.841 + 0.540i)T \)
41 \( 1 + (0.941 + 0.336i)T \)
43 \( 1 + (0.974 - 0.226i)T \)
47 \( 1 + (0.941 - 0.336i)T \)
53 \( 1 + (0.993 - 0.113i)T \)
59 \( 1 + (0.941 - 0.336i)T \)
61 \( 1 + (-0.959 - 0.281i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (-0.985 - 0.170i)T \)
73 \( 1 + (0.993 + 0.113i)T \)
79 \( 1 + (-0.921 + 0.389i)T \)
83 \( 1 + (0.198 - 0.980i)T \)
89 \( 1 + (-0.985 + 0.170i)T \)
97 \( 1 + (-0.921 - 0.389i)T \)
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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

−18.30794667812943328352727352431, −17.60844081055465437391049129812, −16.73185545458485414011464971857, −16.06258469252564685203737823503, −15.49803086527546952854794113831, −14.75287176427526109364413412081, −13.881506828741452279583052661139, −13.39097049196512153457239067624, −12.59814952706550603930971699590, −12.14661335163789101195448032617, −11.3603655265488452384918593059, −10.79997276011025161496518892767, −10.24413236211524726252194598955, −9.53057875012635022487474848274, −7.82789251578854855623937367631, −7.42469296813511066152990303556, −6.90325469748254123046790518452, −5.91992810235517012155941208885, −5.65463511889679153309552389512, −4.40113909789867369404627679696, −3.95334049381497295196658038614, −2.88007147300647978214957526364, −2.449277125898651077079121816058, −1.06216630170705213149499143927, −0.29131005796200329235409998319, 1.280282192720256854272435824454, 2.412267289224083149036790479740, 3.49824172741156418325273708783, 4.06595303856939318756700817708, 4.61424881960389684174526202755, 5.59140914891210646237226226662, 5.90565801224707115626629806120, 6.7478976496499443449732572149, 7.56511060572316580817610454572, 8.472211190387433576915656734101, 9.18822841422965218252280055101, 10.03944685885042248076791776575, 10.86310242143280526893345233382, 11.71884618704238474994418881599, 12.291454487046120740908477186381, 12.456217187174832538291262842, 13.31381707188241933767912548038, 14.55201189842622188964808533379, 14.811739610349688288092742121842, 15.759307181078429815654310134789, 16.23101751150275625411288840334, 16.65795183352829197318406621515, 17.11793813493491987171489979972, 18.30810798836300238467759037358, 19.02486720383564251315791585671

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