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Properties

Label 1-1312-1312.1123-r1-0-0
Degree $1$
Conductor $1312$
Sign $-0.882 - 0.471i$
Analytic cond. $140.993$
Root an. cond. $140.993$
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  + (−0.707 − 0.707i)3-s + (−0.891 − 0.453i)5-s + (−0.587 − 0.809i)7-s + i·9-s + (0.453 + 0.891i)11-s + (0.156 + 0.987i)13-s + (0.309 + 0.951i)15-s + (−0.309 + 0.951i)17-s + (−0.987 − 0.156i)19-s + (−0.156 + 0.987i)21-s + (−0.587 + 0.809i)23-s + (0.587 + 0.809i)25-s + (0.707 − 0.707i)27-s + (−0.453 + 0.891i)29-s + (−0.309 + 0.951i)31-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (−0.891 − 0.453i)5-s + (−0.587 − 0.809i)7-s + i·9-s + (0.453 + 0.891i)11-s + (0.156 + 0.987i)13-s + (0.309 + 0.951i)15-s + (−0.309 + 0.951i)17-s + (−0.987 − 0.156i)19-s + (−0.156 + 0.987i)21-s + (−0.587 + 0.809i)23-s + (0.587 + 0.809i)25-s + (0.707 − 0.707i)27-s + (−0.453 + 0.891i)29-s + (−0.309 + 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1312\)    =    \(2^{5} \cdot 41\)
Sign: $-0.882 - 0.471i$
Analytic conductor: \(140.993\)
Root analytic conductor: \(140.993\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1312} (1123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1312,\ (1:\ ),\ -0.882 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005628233486 + 0.02248875313i\)
\(L(\frac12)\) \(\approx\) \(0.005628233486 + 0.02248875313i\)
\(L(1)\) \(\approx\) \(0.5603566052 - 0.04585977342i\)
\(L(1)\) \(\approx\) \(0.5603566052 - 0.04585977342i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.891 - 0.453i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 + (0.453 + 0.891i)T \)
13 \( 1 + (0.156 + 0.987i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.987 - 0.156i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (-0.453 + 0.891i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.891 - 0.453i)T \)
43 \( 1 + (0.987 - 0.156i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.891 + 0.453i)T \)
59 \( 1 + (-0.987 + 0.156i)T \)
61 \( 1 + (-0.987 - 0.156i)T \)
67 \( 1 + (0.453 - 0.891i)T \)
71 \( 1 + (0.951 + 0.309i)T \)
73 \( 1 + iT \)
79 \( 1 + T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (0.587 + 0.809i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−20.38773578479464313365171838143, −19.50611849440932714920772349433, −18.73538221562050876553672737041, −18.18610192656400558467964217778, −17.14881607751094356941370375926, −16.33484660504482092070370193519, −15.77947279740279559524780868181, −15.17637308227391838714646713853, −14.49877756886374201948795813841, −13.26589564774969285946388477869, −12.32120756164488201531685200610, −11.7493821712640824060688895251, −11.01033461887248681559533999587, −10.36119979882453965870261347725, −9.37226650709082318320285315668, −8.62304340210718512725644390526, −7.72609644552093974229468835441, −6.42709884939000581291653663557, −6.10984296412851318563841332577, −5.05337468166144979564904299814, −4.0508793574080289324897864589, −3.35596098513433741950399017648, −2.48369013486037336536081453082, −0.551670521399444705077962573, −0.009447197388288082429834952779, 1.26697627741433877906001788303, 1.97194926060802487948153243212, 3.66952862343702002477706955420, 4.22256018074392378154464213643, 5.126636244224672434207020059943, 6.36029609977616942994876410563, 6.93693665203268291082145151719, 7.559582373952928673354983285040, 8.55555261520836434391747044697, 9.42939315725109486281602234237, 10.633171448559348385556903577397, 11.077520437478236264866211520420, 12.261076041247868406325346103258, 12.42700607638048557169489150437, 13.34764217447994623163706519416, 14.17007302335175332285784383956, 15.21368632133174272903090958672, 16.06955594559020139009362359635, 16.77622092258124220769593048964, 17.232504077914616530205912948695, 18.121683190256178456150385247120, 19.154190982384008269182492807935, 19.61896226031104972464955907372, 20.078095541926463228132350418834, 21.296149017007307292414825298629

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