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Properties

Label 1-3751-3751.2113-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.999 + 0.0289i$
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  + (0.841 + 0.540i)2-s + (−0.5 − 0.866i)3-s + (0.415 + 0.909i)4-s + (0.928 + 0.371i)5-s + (0.0475 − 0.998i)6-s + (−0.327 + 0.945i)7-s + (−0.142 + 0.989i)8-s + (−0.5 + 0.866i)9-s + (0.580 + 0.814i)10-s + (0.580 − 0.814i)12-s + (−0.995 − 0.0950i)13-s + (−0.786 + 0.618i)14-s + (−0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (0.723 + 0.690i)17-s + (−0.888 + 0.458i)18-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (−0.5 − 0.866i)3-s + (0.415 + 0.909i)4-s + (0.928 + 0.371i)5-s + (0.0475 − 0.998i)6-s + (−0.327 + 0.945i)7-s + (−0.142 + 0.989i)8-s + (−0.5 + 0.866i)9-s + (0.580 + 0.814i)10-s + (0.580 − 0.814i)12-s + (−0.995 − 0.0950i)13-s + (−0.786 + 0.618i)14-s + (−0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (0.723 + 0.690i)17-s + (−0.888 + 0.458i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.999 + 0.0289i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (2113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.999 + 0.0289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02434751037 + 1.680166936i\)
\(L(\frac12)\) \(\approx\) \(0.02434751037 + 1.680166936i\)
\(L(1)\) \(\approx\) \(1.191640943 + 0.6884412161i\)
\(L(1)\) \(\approx\) \(1.191640943 + 0.6884412161i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.928 + 0.371i)T \)
7 \( 1 + (-0.327 + 0.945i)T \)
13 \( 1 + (-0.995 - 0.0950i)T \)
17 \( 1 + (0.723 + 0.690i)T \)
19 \( 1 + (0.235 + 0.971i)T \)
23 \( 1 + (-0.654 + 0.755i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
37 \( 1 + (-0.995 + 0.0950i)T \)
41 \( 1 + (-0.888 + 0.458i)T \)
43 \( 1 + (0.928 - 0.371i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (0.981 - 0.189i)T \)
59 \( 1 + (-0.888 - 0.458i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (0.723 - 0.690i)T \)
73 \( 1 + (-0.327 - 0.945i)T \)
79 \( 1 + (-0.786 + 0.618i)T \)
83 \( 1 + (0.981 - 0.189i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−18.20318355711302381255477843682, −17.41725926044517476758578748862, −16.78207218268831323153898706435, −16.26955030267420748427132774666, −15.526653991572576239264977082674, −14.63102688028145366214770652571, −14.096789357741444103375322090326, −13.56654196830366493271870346606, −12.64400944724196502938323821220, −12.16125588555602944808964727296, −11.32142398843833543871819513052, −10.51991512362185990585874832945, −10.10869255379354140309462119515, −9.51903119020535781637850055582, −8.90702317068617169509732146591, −7.32707043735958953445808484245, −6.781008008599101437865905453083, −5.832673373291525361563820940772, −5.30387232312706957614103575108, −4.62648605695880164643120785308, −4.03820299195208264429519474281, −3.07969095001693219789841306028, −2.409063114007652381819853826861, −1.23221647249583430703795710090, −0.34635591719601949438338925498, 1.66725897136938451239708763708, 2.133723040225030897961425447031, 2.95382620819476751656512574427, 3.79175394609169632840798630654, 5.22423966012305705128371790536, 5.48828124878027080999376449854, 6.06226365551640634886730681049, 6.73788347864198324394197643335, 7.51333889945608555074794430151, 8.10397294441957822644252620565, 9.0737121801883567178630126772, 9.979775081178766309640010430820, 10.72975673202663508525617845397, 11.85476373141712480037656821611, 12.111836493664156245931237935185, 12.791493245816368266756783284559, 13.44234936985474747164745456087, 14.117916485728739941432239861657, 14.71136839465260799289184033813, 15.366764885078262654108742749234, 16.32872025745049619886074521394, 16.98316241285424805916912892341, 17.40306052664167946402760787065, 18.19333452110182160419312909680, 18.77914942466578169235779640983

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