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Properties

Label 1-740-740.459-r0-0-0
Degree $1$
Conductor $740$
Sign $0.379 - 0.925i$
Analytic cond. $3.43654$
Root an. cond. $3.43654$
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  + (−0.766 + 0.642i)3-s + (−0.939 − 0.342i)7-s + (0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.984 + 0.173i)13-s + (0.984 + 0.173i)17-s + (0.642 + 0.766i)19-s + (0.939 − 0.342i)21-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)27-s + (−0.866 − 0.5i)29-s − i·31-s + (−0.173 − 0.984i)33-s + (0.642 − 0.766i)39-s + (−0.173 − 0.984i)41-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (−0.939 − 0.342i)7-s + (0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.984 + 0.173i)13-s + (0.984 + 0.173i)17-s + (0.642 + 0.766i)19-s + (0.939 − 0.342i)21-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)27-s + (−0.866 − 0.5i)29-s − i·31-s + (−0.173 − 0.984i)33-s + (0.642 − 0.766i)39-s + (−0.173 − 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(3.43654\)
Root analytic conductor: \(3.43654\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 740,\ (0:\ ),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3805085779 - 0.2551512424i\)
\(L(\frac12)\) \(\approx\) \(0.3805085779 - 0.2551512424i\)
\(L(1)\) \(\approx\) \(0.6237727290 + 0.06450112604i\)
\(L(1)\) \(\approx\) \(0.6237727290 + 0.06450112604i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (0.342 - 0.939i)T \)
83 \( 1 + (0.173 - 0.984i)T \)
89 \( 1 + (-0.342 - 0.939i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−22.60835515930219675037145190286, −22.00080209640723417693769411597, −21.301226324529512713117025276040, −19.87419069142193243012419025453, −19.40509896194390664315199792374, −18.427701233841802589299241452354, −18.02765290170518006953792164524, −16.70567336904025233993484026004, −16.424291902620931120094662230415, −15.49105100375400413393815186736, −14.27692598593728889961806789873, −13.435621895830377388562948613023, −12.60981302149399372312041168838, −12.05860144021142526717099152718, −11.093616671372347087311085711801, −10.19122768660285331999383515142, −9.364627008314018199238890330891, −8.11043204629873593362099605573, −7.33211927160417182567912637227, −6.40020481671372623644693065380, −5.60832177286475627316693534851, −4.86682604450030438593166880797, −3.2745747912412164502744691366, −2.48917086768761386895845018, −1.00447296564445147406843746590, 0.2832173096051617890264876256, 1.98492192129756423706793500079, 3.39899866447145029245315165193, 4.1183201696772976561133363416, 5.26868311595961143633893284082, 5.9068222897472085104141775634, 7.0630322694922281235963998684, 7.73885895877651458684152668909, 9.33683798412951477765202763634, 9.97541459373026274066919633488, 10.33049640857923280593731313238, 11.73082333635335795572234197493, 12.24224004198375860436965808447, 13.10066906355784027131401684910, 14.25452047105888246504077622755, 15.17002902434725939189682344517, 15.86269393586173433683670315891, 16.759118752601551011930158923357, 17.169292275807635574382286988823, 18.25530760494049864998514120479, 19.0156041491345923344941456930, 20.128610269249596207212980811434, 20.69097096042691471794012372887, 21.690273793127680961101460426384, 22.431317280489186878310387839609

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