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Properties

Label 1-1049-1049.106-r0-0-0
Degree $1$
Conductor $1049$
Sign $0.989 - 0.141i$
Analytic cond. $4.87153$
Root an. cond. $4.87153$
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  + (0.971 + 0.237i)2-s + (−0.0838 − 0.996i)3-s + (0.887 + 0.461i)4-s + (0.363 + 0.931i)5-s + (0.155 − 0.987i)6-s + (0.554 − 0.832i)7-s + (0.752 + 0.658i)8-s + (−0.985 + 0.167i)9-s + (0.131 + 0.991i)10-s + (0.864 − 0.503i)11-s + (0.385 − 0.922i)12-s + (0.685 + 0.727i)13-s + (0.736 − 0.676i)14-s + (0.897 − 0.440i)15-s + (0.574 + 0.818i)16-s + (−0.649 − 0.760i)17-s + ⋯
L(s)  = 1  + (0.971 + 0.237i)2-s + (−0.0838 − 0.996i)3-s + (0.887 + 0.461i)4-s + (0.363 + 0.931i)5-s + (0.155 − 0.987i)6-s + (0.554 − 0.832i)7-s + (0.752 + 0.658i)8-s + (−0.985 + 0.167i)9-s + (0.131 + 0.991i)10-s + (0.864 − 0.503i)11-s + (0.385 − 0.922i)12-s + (0.685 + 0.727i)13-s + (0.736 − 0.676i)14-s + (0.897 − 0.440i)15-s + (0.574 + 0.818i)16-s + (−0.649 − 0.760i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $0.989 - 0.141i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ 0.989 - 0.141i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.374154264 - 0.2403080471i\)
\(L(\frac12)\) \(\approx\) \(3.374154264 - 0.2403080471i\)
\(L(1)\) \(\approx\) \(2.171652885 - 0.1042754548i\)
\(L(1)\) \(\approx\) \(2.171652885 - 0.1042754548i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (0.971 + 0.237i)T \)
3 \( 1 + (-0.0838 - 0.996i)T \)
5 \( 1 + (0.363 + 0.931i)T \)
7 \( 1 + (0.554 - 0.832i)T \)
11 \( 1 + (0.864 - 0.503i)T \)
13 \( 1 + (0.685 + 0.727i)T \)
17 \( 1 + (-0.649 - 0.760i)T \)
19 \( 1 + (0.981 - 0.190i)T \)
23 \( 1 + (0.202 + 0.979i)T \)
29 \( 1 + (-0.631 - 0.775i)T \)
31 \( 1 + (-0.995 - 0.0957i)T \)
37 \( 1 + (0.249 + 0.968i)T \)
41 \( 1 + (-0.981 + 0.190i)T \)
43 \( 1 + (0.225 - 0.974i)T \)
47 \( 1 + (0.985 - 0.167i)T \)
53 \( 1 + (-0.593 - 0.804i)T \)
59 \( 1 + (-0.736 - 0.676i)T \)
61 \( 1 + (0.450 + 0.892i)T \)
67 \( 1 + (0.918 + 0.396i)T \)
71 \( 1 + (0.155 + 0.987i)T \)
73 \( 1 + (-0.472 + 0.881i)T \)
79 \( 1 + (0.450 - 0.892i)T \)
83 \( 1 + (0.513 - 0.857i)T \)
89 \( 1 + (0.554 + 0.832i)T \)
97 \( 1 + (-0.825 - 0.564i)T \)
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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

−21.69832834235781270023739573140, −20.77119682696424500311295957952, −20.32058513938042570438270182677, −19.79592303002844659167837306680, −18.3793501468830280356774128625, −17.46655982376719063020887845400, −16.60810710535726760402541328892, −15.925757347785182152546225899466, −15.16485380613212881529152226774, −14.581280972057221455486817806993, −13.75948051939935464071364414904, −12.631351364119229208972838535878, −12.22713469645087677272556526091, −11.18560983925481338571890590653, −10.62963760880532600796447490728, −9.44459373136082188027844543571, −8.935360431581270080122679880740, −7.89898773754229866198810457825, −6.3757544506646745702693489024, −5.641698092028269455721779587, −5.04086517498639409918831464080, −4.24812019671487822079967948823, −3.438661737778765007405357672568, −2.2343354698137822868342272571, −1.28513433714285265100125181933, 1.30438062654417628532779985814, 2.08596089811010909725755503072, 3.23456720117758570707106019746, 3.93633796600646884200713213016, 5.25208473936710273003350763797, 6.057222838801704969572095577591, 6.99897923569488766009919617977, 7.1547589709639864276332100900, 8.245543399960847277092441075352, 9.4321179627712007247774340384, 10.850877681030095441746757082223, 11.49409424788115504654969731063, 11.72170075503114869870124714492, 13.31176836134870244502368136904, 13.63914472926204507564374099537, 14.13400627193116050257112727169, 14.88847167709773135213815511730, 15.977874665184269937499852973046, 16.98179492696103069082204409503, 17.46442693711731016919561359993, 18.42035647619843941496048645287, 19.1571302174794867938668676393, 20.15512415216074523211375081091, 20.656905613795736644499832202118, 21.926833564540910210996454168964

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