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Properties

Label 1-3751-3751.2712-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.999 + 0.0177i$
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  + (0.564 + 0.825i)2-s + (0.809 − 0.587i)3-s + (−0.362 + 0.931i)4-s + (0.0855 + 0.996i)5-s + (0.941 + 0.336i)6-s + (0.870 + 0.491i)7-s + (−0.974 + 0.226i)8-s + (0.309 − 0.951i)9-s + (−0.774 + 0.633i)10-s + (0.254 + 0.967i)12-s + (−0.998 − 0.0570i)13-s + (0.0855 + 0.996i)14-s + (0.654 + 0.755i)15-s + (−0.736 − 0.676i)16-s + (−0.142 + 0.989i)17-s + (0.959 − 0.281i)18-s + ⋯
L(s)  = 1  + (0.564 + 0.825i)2-s + (0.809 − 0.587i)3-s + (−0.362 + 0.931i)4-s + (0.0855 + 0.996i)5-s + (0.941 + 0.336i)6-s + (0.870 + 0.491i)7-s + (−0.974 + 0.226i)8-s + (0.309 − 0.951i)9-s + (−0.774 + 0.633i)10-s + (0.254 + 0.967i)12-s + (−0.998 − 0.0570i)13-s + (0.0855 + 0.996i)14-s + (0.654 + 0.755i)15-s + (−0.736 − 0.676i)16-s + (−0.142 + 0.989i)17-s + (0.959 − 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.999 + 0.0177i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (2712, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.999 + 0.0177i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02086880194 + 2.354833815i\)
\(L(\frac12)\) \(\approx\) \(0.02086880194 + 2.354833815i\)
\(L(1)\) \(\approx\) \(1.267218629 + 1.048794143i\)
\(L(1)\) \(\approx\) \(1.267218629 + 1.048794143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.564 + 0.825i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.0855 + 0.996i)T \)
7 \( 1 + (0.870 + 0.491i)T \)
13 \( 1 + (-0.998 - 0.0570i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (-0.198 - 0.980i)T \)
29 \( 1 + (-0.466 + 0.884i)T \)
37 \( 1 + (-0.774 + 0.633i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (0.516 + 0.856i)T \)
47 \( 1 + (0.941 - 0.336i)T \)
53 \( 1 + (-0.993 + 0.113i)T \)
59 \( 1 + (-0.959 - 0.281i)T \)
61 \( 1 + (0.610 + 0.791i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.198 + 0.980i)T \)
79 \( 1 + (-0.921 + 0.389i)T \)
83 \( 1 + (-0.870 - 0.491i)T \)
89 \( 1 + (-0.696 + 0.717i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−18.427593304174322660964313045431, −17.47048942337631212766671819079, −17.07490528471134144076663230924, −15.811486226152169812449618972272, −15.58113568463934779925234571268, −14.59422785014159507806772904450, −14.01118400467566012973307753684, −13.569084525050272179880947178102, −12.87408928488602004337038838417, −12.00880024289697834147566018433, −11.374671272008419412488554393736, −10.69009884612168450914035233617, −9.77095045812524704177719497273, −9.34299027563035756346745055788, −8.77338670406091068231990540347, −7.746433660663574425757396493969, −7.21218150789675183844894438856, −5.66204109399555259262950936134, −5.08844091489419473226740374803, −4.48567341827418211256982059202, −4.04404274020112762876185300096, −2.97300461263591174820878871270, −2.21262833242203646730105127716, −1.53777765256642347551258931838, −0.432274297482015535451491924130, 1.5710966687629029849812627528, 2.38242741514159232184171283514, 2.987632982494322937160076043386, 3.866893599853111050224514276909, 4.59162209191609368084603284662, 5.6964925311305077051536835018, 6.204777443556196707881783503501, 7.099829497436965769870762303018, 7.57898135159348376012546539100, 8.23204262235813337657551443724, 8.837900367336130872172830762827, 9.75120829651417503364304362847, 10.64226669938322421437636860773, 11.59077036758140777617842133579, 12.449528736106902025578451420206, 12.652973872406892083334258925205, 13.884968355647719003174049059237, 14.235819709132644418963666231377, 14.84886499072653663346922918273, 15.07411098723314141663497138559, 15.98893006676046683157104825888, 17.05143751133884384324349811048, 17.6029211319733398277693891797, 18.27057155126122679861272010611, 18.74879710271112208370369879014

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