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Properties

Label 1-1027-1027.114-r1-0-0
Degree $1$
Conductor $1027$
Sign $0.649 + 0.759i$
Analytic cond. $110.366$
Root an. cond. $110.366$
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  + (−0.120 − 0.992i)2-s + (0.354 − 0.935i)3-s + (−0.970 + 0.239i)4-s + (0.845 − 0.534i)5-s + (−0.970 − 0.239i)6-s + (−0.354 + 0.935i)7-s + (0.354 + 0.935i)8-s + (−0.748 − 0.663i)9-s + (−0.632 − 0.774i)10-s + (0.845 + 0.534i)11-s + (−0.120 + 0.992i)12-s + (0.970 + 0.239i)14-s + (−0.200 − 0.979i)15-s + (0.885 − 0.464i)16-s + (−0.692 − 0.721i)17-s + (−0.568 + 0.822i)18-s + ⋯
L(s)  = 1  + (−0.120 − 0.992i)2-s + (0.354 − 0.935i)3-s + (−0.970 + 0.239i)4-s + (0.845 − 0.534i)5-s + (−0.970 − 0.239i)6-s + (−0.354 + 0.935i)7-s + (0.354 + 0.935i)8-s + (−0.748 − 0.663i)9-s + (−0.632 − 0.774i)10-s + (0.845 + 0.534i)11-s + (−0.120 + 0.992i)12-s + (0.970 + 0.239i)14-s + (−0.200 − 0.979i)15-s + (0.885 − 0.464i)16-s + (−0.692 − 0.721i)17-s + (−0.568 + 0.822i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.649 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.649 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $0.649 + 0.759i$
Analytic conductor: \(110.366\)
Root analytic conductor: \(110.366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (1:\ ),\ 0.649 + 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1575032128 + 0.07254392416i\)
\(L(\frac12)\) \(\approx\) \(0.1575032128 + 0.07254392416i\)
\(L(1)\) \(\approx\) \(0.6986135676 - 0.6272661636i\)
\(L(1)\) \(\approx\) \(0.6986135676 - 0.6272661636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.120 - 0.992i)T \)
3 \( 1 + (0.354 - 0.935i)T \)
5 \( 1 + (0.845 - 0.534i)T \)
7 \( 1 + (-0.354 + 0.935i)T \)
11 \( 1 + (0.845 + 0.534i)T \)
17 \( 1 + (-0.692 - 0.721i)T \)
19 \( 1 + (-0.568 + 0.822i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.948 - 0.316i)T \)
31 \( 1 + (0.919 + 0.391i)T \)
37 \( 1 + (0.568 - 0.822i)T \)
41 \( 1 + (-0.845 + 0.534i)T \)
43 \( 1 + (0.845 - 0.534i)T \)
47 \( 1 + (-0.996 + 0.0804i)T \)
53 \( 1 + (-0.987 - 0.160i)T \)
59 \( 1 + (0.692 - 0.721i)T \)
61 \( 1 + (0.996 + 0.0804i)T \)
67 \( 1 + (-0.799 - 0.600i)T \)
71 \( 1 + (-0.632 + 0.774i)T \)
73 \( 1 + (-0.692 + 0.721i)T \)
83 \( 1 + (-0.278 + 0.960i)T \)
89 \( 1 + (0.632 + 0.774i)T \)
97 \( 1 + (0.996 + 0.0804i)T \)
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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

−21.6159690116860357644236347332, −20.56314165294587145319880108681, −19.52219536446893505600786913907, −19.10646083578008684070150123947, −17.702953497729120626965553544539, −17.24817757949570493357782292390, −16.639713023781531809623942158285, −15.7928690141521530030033773293, −14.91243137172803692006569272402, −14.4184861992426156281593812192, −13.43376530934087628995951152105, −13.30089890390888837445943717763, −11.370945176415823974068369335973, −10.56454844092074881481718191266, −9.85238576650305914317658707158, −9.2149584588180283944763560113, −8.43421361731533445758856514162, −7.349932472415389164031310750631, −6.42935528488335836576697254492, −5.876202980693142786129965798372, −4.67762481855519775607292939757, −3.942289549715828770867150053443, −3.08729072992489482549677434383, −1.570216201631044240635040995112, −0.03519086888016947900543306777, 1.17392595026890562037265452209, 2.10047151710836225900581502639, 2.506396758923357363955428656716, 3.786717054600191815527280597572, 4.8956590580506719535835581544, 5.94320968148969542901320271316, 6.656840515685154601193478945160, 8.046652807569547852106383680357, 8.791298940564861836910307217172, 9.35587399110426092382822797254, 10.06396578059647969150580199277, 11.4140319273635128195330295175, 12.11598607609717296630619719039, 12.71296464396772967336954293600, 13.26952118769361008589418621181, 14.23054132627320148269288166749, 14.747865350969451029085584436553, 16.22327316500114995347707577200, 17.23949462350279836905647136324, 17.783648979081904824009836201083, 18.52631719546157576677342592108, 19.13047769064694410848656470377, 20.00515233217614457696206965232, 20.57390678081779733568295661014, 21.3053058105372689830055388770

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