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Properties

Label 1-1027-1027.128-r1-0-0
Degree $1$
Conductor $1027$
Sign $0.598 - 0.801i$
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  + (0.992 + 0.120i)2-s + (−0.354 − 0.935i)3-s + (0.970 + 0.239i)4-s + (−0.999 − 0.0402i)5-s + (−0.239 − 0.970i)6-s + (−0.935 + 0.354i)7-s + (0.935 + 0.354i)8-s + (−0.748 + 0.663i)9-s + (−0.987 − 0.160i)10-s + (−0.999 + 0.0402i)11-s + (−0.120 − 0.992i)12-s + (−0.970 + 0.239i)14-s + (0.316 + 0.948i)15-s + (0.885 + 0.464i)16-s + (−0.278 − 0.960i)17-s + (−0.822 + 0.568i)18-s + ⋯
L(s)  = 1  + (0.992 + 0.120i)2-s + (−0.354 − 0.935i)3-s + (0.970 + 0.239i)4-s + (−0.999 − 0.0402i)5-s + (−0.239 − 0.970i)6-s + (−0.935 + 0.354i)7-s + (0.935 + 0.354i)8-s + (−0.748 + 0.663i)9-s + (−0.987 − 0.160i)10-s + (−0.999 + 0.0402i)11-s + (−0.120 − 0.992i)12-s + (−0.970 + 0.239i)14-s + (0.316 + 0.948i)15-s + (0.885 + 0.464i)16-s + (−0.278 − 0.960i)17-s + (−0.822 + 0.568i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $0.598 - 0.801i$
Analytic conductor: \(110.366\)
Root analytic conductor: \(110.366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (1:\ ),\ 0.598 - 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.408079135 - 0.7060022522i\)
\(L(\frac12)\) \(\approx\) \(1.408079135 - 0.7060022522i\)
\(L(1)\) \(\approx\) \(1.122974910 - 0.1634110539i\)
\(L(1)\) \(\approx\) \(1.122974910 - 0.1634110539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.992 + 0.120i)T \)
3 \( 1 + (-0.354 - 0.935i)T \)
5 \( 1 + (-0.999 - 0.0402i)T \)
7 \( 1 + (-0.935 + 0.354i)T \)
11 \( 1 + (-0.999 + 0.0402i)T \)
17 \( 1 + (-0.278 - 0.960i)T \)
19 \( 1 + (-0.822 + 0.568i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.200 + 0.979i)T \)
31 \( 1 + (-0.600 + 0.799i)T \)
37 \( 1 + (0.822 - 0.568i)T \)
41 \( 1 + (-0.999 - 0.0402i)T \)
43 \( 1 + (0.0402 - 0.999i)T \)
47 \( 1 + (0.903 - 0.428i)T \)
53 \( 1 + (-0.632 - 0.774i)T \)
59 \( 1 + (-0.960 - 0.278i)T \)
61 \( 1 + (0.428 - 0.903i)T \)
67 \( 1 + (0.391 - 0.919i)T \)
71 \( 1 + (0.160 + 0.987i)T \)
73 \( 1 + (-0.960 - 0.278i)T \)
83 \( 1 + (0.721 + 0.692i)T \)
89 \( 1 + (0.160 - 0.987i)T \)
97 \( 1 + (0.903 + 0.428i)T \)
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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

−21.72446029724283474175910723155, −20.7254241780493175849268826264, −20.23769556336206524483423783442, −19.39942397821591556864071859488, −18.713625866108110789118018126971, −17.143231267353214242847269584625, −16.541218535575923417822598880232, −15.826487498279812493326106006095, −15.19306670832004283926832603518, −14.77667018097653226129098612081, −13.388469206374837021722268559552, −12.823706229556703751457000185759, −12.004276243680312493006158420281, −10.955146036391340485835175071344, −10.69707949144993260221058941116, −9.746813795605028904910757667442, −8.52189527010646149086724271849, −7.521690001465388157845147300637, −6.48323383286993316127751672856, −5.84458889188333185989770160224, −4.57635114835387824313232337654, −4.23727970678179113369311493168, −3.26465992060980837665010057451, −2.55949661912445653645412869370, −0.588078369659746961999095599849, 0.39366639099307503497959975993, 1.91764390117197301368599824337, 2.90891419431335132579124701522, 3.59917876130814953051154912033, 4.944476670836131227887520947269, 5.53370254136286283049198809689, 6.627180242476028592835834734595, 7.19299790104606566591653704327, 7.92058644262388032453327064399, 8.91506447151611678896411253683, 10.47352101606527019529500100953, 11.17321681348397373574053950642, 12.00140893877871862040488194565, 12.65721325860991660371063043405, 13.088924393268464508110313378866, 14.008036532277304170729864832861, 15.001486167475327934510537471649, 15.82093641810076424453980635102, 16.29058420148452392286523240789, 17.175173424957833062413266361279, 18.46761884134645442904586684792, 18.91065978709123653809647177328, 19.86460012681373304036817642902, 20.30909222251439221536777113664, 21.514360187753869724639534377108

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