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Properties

Label 1-3751-3751.1040-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.261 + 0.965i$
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  + (0.142 + 0.989i)2-s + (−0.913 + 0.406i)3-s + (−0.959 + 0.281i)4-s + (0.861 + 0.508i)5-s + (−0.532 − 0.846i)6-s + (0.888 − 0.458i)7-s + (−0.415 − 0.909i)8-s + (0.669 − 0.743i)9-s + (−0.380 + 0.924i)10-s + (0.761 − 0.647i)12-s + (0.879 − 0.475i)13-s + (0.580 + 0.814i)14-s + (−0.993 − 0.113i)15-s + (0.841 − 0.540i)16-s + (−0.683 + 0.730i)17-s + (0.830 + 0.556i)18-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)2-s + (−0.913 + 0.406i)3-s + (−0.959 + 0.281i)4-s + (0.861 + 0.508i)5-s + (−0.532 − 0.846i)6-s + (0.888 − 0.458i)7-s + (−0.415 − 0.909i)8-s + (0.669 − 0.743i)9-s + (−0.380 + 0.924i)10-s + (0.761 − 0.647i)12-s + (0.879 − 0.475i)13-s + (0.580 + 0.814i)14-s + (−0.993 − 0.113i)15-s + (0.841 − 0.540i)16-s + (−0.683 + 0.730i)17-s + (0.830 + 0.556i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.261 + 0.965i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1040, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.261 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.112211956 + 1.454131374i\)
\(L(\frac12)\) \(\approx\) \(1.112211956 + 1.454131374i\)
\(L(1)\) \(\approx\) \(0.8884959581 + 0.6710816624i\)
\(L(1)\) \(\approx\) \(0.8884959581 + 0.6710816624i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.142 + 0.989i)T \)
3 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (0.861 + 0.508i)T \)
7 \( 1 + (0.888 - 0.458i)T \)
13 \( 1 + (0.879 - 0.475i)T \)
17 \( 1 + (-0.683 + 0.730i)T \)
19 \( 1 + (0.999 - 0.0190i)T \)
23 \( 1 + (0.254 + 0.967i)T \)
29 \( 1 + (0.0855 - 0.996i)T \)
37 \( 1 + (0.432 + 0.901i)T \)
41 \( 1 + (-0.345 - 0.938i)T \)
43 \( 1 + (0.749 + 0.662i)T \)
47 \( 1 + (0.696 - 0.717i)T \)
53 \( 1 + (-0.964 + 0.263i)T \)
59 \( 1 + (0.345 - 0.938i)T \)
61 \( 1 + (0.696 - 0.717i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (-0.905 + 0.424i)T \)
73 \( 1 + (-0.888 - 0.458i)T \)
79 \( 1 + (-0.595 + 0.803i)T \)
83 \( 1 + (0.964 - 0.263i)T \)
89 \( 1 + (-0.516 + 0.856i)T \)
97 \( 1 + (0.198 - 0.980i)T \)
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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

−18.39039374503594063025907621768, −17.81894780867332056898622363593, −17.464053976115831862310324646744, −16.42874501847456672144525232311, −15.95023887179017467091081241622, −14.661507412098682960053442693105, −14.03679546015020079005336204707, −13.40627408882118730906347385784, −12.8369139787448973566220674848, −12.05419546056269840943952828439, −11.58557996712466083429495562472, −10.85078938363538218707213030191, −10.38796067981859199347172681632, −9.25086341270121140096369769324, −8.94529145387036517763625426523, −8.02448174819713375617899742593, −6.987862316194773126993426112358, −5.98392250505173065429440017859, −5.528670893980647205306901780187, −4.72841780609296439096937397026, −4.34694111677063564019236733233, −2.90564928152829116403850251412, −2.1100850558061333125095976327, −1.41726425916594692103118515502, −0.81181748364799414631471872010, 0.86249949595747894774864957635, 1.67246640843138196828964725299, 3.16615261982969248628778413667, 3.96512394848457069759826777165, 4.695959980699226324559209050994, 5.53596023267076031973793557614, 5.875874474621487000248129466227, 6.67863102521878910626760782685, 7.34430476852656070158818119085, 8.14825397060762935509033922978, 9.041020133222370310958589291263, 9.794794130631648105311113775892, 10.41609868078522837579848149314, 11.13891717051449352552794423439, 11.7660086834491794415344434054, 12.913953983356477182040251994232, 13.44229449341217583758019283384, 14.04617564167072247312919317, 14.858367443712964619345742222722, 15.4703433255814562189851798690, 15.979763615851339982383380089890, 17.03146219898794425582920278557, 17.33980234526688935421930222490, 17.84981611729925570957727692569, 18.34442701234508576118758390691

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