Properties

Label 1-740-740.79-r0-0-0
Degree 11
Conductor 740740
Sign 0.379+0.925i0.379 + 0.925i
Analytic cond. 3.436543.43654
Root an. cond. 3.436543.43654
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(740s/2ΓR(s)L(s)=((0.379+0.925i)Λ(1s)\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}
Λ(s)=(740s/2ΓR(s)L(s)=((0.379+0.925i)Λ(1s)\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: 11
Conductor: 740740    =    225372^{2} \cdot 5 \cdot 37
Sign: 0.379+0.925i0.379 + 0.925i
Analytic conductor: 3.436543.43654
Root analytic conductor: 3.436543.43654
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ740(79,)\chi_{740} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 740, (0: ), 0.379+0.925i)(1,\ 740,\ (0:\ ),\ 0.379 + 0.925i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.3805085779+0.2551512424i0.3805085779 + 0.2551512424i
L(12)L(\frac12) \approx 0.3805085779+0.2551512424i0.3805085779 + 0.2551512424i
L(1)L(1) \approx 0.62377272900.06450112604i0.6237727290 - 0.06450112604i
L(1)L(1) \approx 0.62377272900.06450112604i0.6237727290 - 0.06450112604i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
37 1 1
good3 1+(0.7660.642i)T 1 + (-0.766 - 0.642i)T
7 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
11 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
13 1+(0.9840.173i)T 1 + (-0.984 - 0.173i)T
17 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
19 1+(0.6420.766i)T 1 + (0.642 - 0.766i)T
23 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
29 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
31 1iT 1 - iT
41 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
43 1iT 1 - iT
47 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
53 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
59 1+(0.342+0.939i)T 1 + (-0.342 + 0.939i)T
61 1+(0.9840.173i)T 1 + (-0.984 - 0.173i)T
67 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
71 1+(0.7660.642i)T 1 + (-0.766 - 0.642i)T
73 1+T 1 + T
79 1+(0.342+0.939i)T 1 + (0.342 + 0.939i)T
83 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
89 1+(0.342+0.939i)T 1 + (-0.342 + 0.939i)T
97 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

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

Graph of the ZZ-function along the critical line