Properties

Label 1-26e2-676.47-r0-0-0
Degree 11
Conductor 676676
Sign 0.1750.984i-0.175 - 0.984i
Analytic cond. 3.139333.13933
Root an. cond. 3.139333.13933
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  + (−0.885 − 0.464i)3-s + (−0.663 − 0.748i)5-s + (−0.239 − 0.970i)7-s + (0.568 + 0.822i)9-s + (0.822 + 0.568i)11-s + (0.239 + 0.970i)15-s + (0.970 − 0.239i)17-s i·19-s + (−0.239 + 0.970i)21-s + 23-s + (−0.120 + 0.992i)25-s + (−0.120 − 0.992i)27-s + (0.568 + 0.822i)29-s + (0.992 − 0.120i)31-s + (−0.464 − 0.885i)33-s + ⋯
L(s)  = 1  + (−0.885 − 0.464i)3-s + (−0.663 − 0.748i)5-s + (−0.239 − 0.970i)7-s + (0.568 + 0.822i)9-s + (0.822 + 0.568i)11-s + (0.239 + 0.970i)15-s + (0.970 − 0.239i)17-s i·19-s + (−0.239 + 0.970i)21-s + 23-s + (−0.120 + 0.992i)25-s + (−0.120 − 0.992i)27-s + (0.568 + 0.822i)29-s + (0.992 − 0.120i)31-s + (−0.464 − 0.885i)33-s + ⋯

Functional equation

Λ(s)=(676s/2ΓR(s)L(s)=((0.1750.984i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(676s/2ΓR(s)L(s)=((0.1750.984i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 676676    =    221322^{2} \cdot 13^{2}
Sign: 0.1750.984i-0.175 - 0.984i
Analytic conductor: 3.139333.13933
Root analytic conductor: 3.139333.13933
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ676(47,)\chi_{676} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 676, (0: ), 0.1750.984i)(1,\ 676,\ (0:\ ),\ -0.175 - 0.984i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.60033627760.7169553146i0.6003362776 - 0.7169553146i
L(12)L(\frac12) \approx 0.60033627760.7169553146i0.6003362776 - 0.7169553146i
L(1)L(1) \approx 0.73150172830.3223538015i0.7315017283 - 0.3223538015i
L(1)L(1) \approx 0.73150172830.3223538015i0.7315017283 - 0.3223538015i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
13 1 1
good3 1+(0.8850.464i)T 1 + (-0.885 - 0.464i)T
5 1+(0.6630.748i)T 1 + (-0.663 - 0.748i)T
7 1+(0.2390.970i)T 1 + (-0.239 - 0.970i)T
11 1+(0.822+0.568i)T 1 + (0.822 + 0.568i)T
17 1+(0.9700.239i)T 1 + (0.970 - 0.239i)T
19 1iT 1 - iT
23 1+T 1 + T
29 1+(0.568+0.822i)T 1 + (0.568 + 0.822i)T
31 1+(0.9920.120i)T 1 + (0.992 - 0.120i)T
37 1+(0.9920.120i)T 1 + (0.992 - 0.120i)T
41 1+(0.464+0.885i)T 1 + (-0.464 + 0.885i)T
43 1+(0.1200.992i)T 1 + (0.120 - 0.992i)T
47 1+(0.9350.354i)T 1 + (0.935 - 0.354i)T
53 1+(0.970+0.239i)T 1 + (-0.970 + 0.239i)T
59 1+(0.6630.748i)T 1 + (-0.663 - 0.748i)T
61 1+(0.9700.239i)T 1 + (-0.970 - 0.239i)T
67 1+(0.935+0.354i)T 1 + (-0.935 + 0.354i)T
71 1+(0.4640.885i)T 1 + (0.464 - 0.885i)T
73 1+(0.8220.568i)T 1 + (-0.822 - 0.568i)T
79 1+(0.354+0.935i)T 1 + (0.354 + 0.935i)T
83 1+(0.4640.885i)T 1 + (-0.464 - 0.885i)T
89 1iT 1 - iT
97 1+(0.6630.748i)T 1 + (0.663 - 0.748i)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

−22.89067091228411552516001962554, −22.25158057031116773056695991002, −21.51922864055352068663284799702, −20.816727008888775907448069876171, −19.3510257369414957068745343190, −18.93025263000201176238533534357, −18.15519787112302205167726655619, −17.09478222483286349446421813518, −16.39502717898574402153510755713, −15.53644366696405221351536912602, −14.925890381359816333241555516996, −14.08080551331659637010929548905, −12.58031799543151122989725058063, −11.94180833012883884423814431976, −11.37267386812766901347771107948, −10.419327934620607597661985602381, −9.612199669900761504016109355563, −8.57259567315531574274417471051, −7.50741674128456452812125255915, −6.27136204611452497478621764728, −5.97106779166208142084108631110, −4.67311354288821745639293260850, −3.665623282156886884532442277953, −2.85370802692362787602399456012, −1.116257865844979678858322427452, 0.68610962558817533855120446913, 1.418828151152139105436081783889, 3.20418582751835058582267865769, 4.46465510554987139557577397156, 4.89463794164094366507002406926, 6.244103366292967015106434175128, 7.13706640450052574030294240893, 7.688431746905325337670384847395, 8.93911891251075185328287001983, 9.9342688946259410217557586733, 10.918351278450922548549936489769, 11.71881576960149565945341624320, 12.430194058888386441029695196426, 13.16014103481084625199680541080, 14.04401858470561226607716412295, 15.26384035405670190940315898505, 16.15670273040683097517182665061, 17.01040631590041854989270691501, 17.20502713200000745273391808555, 18.42055167758261468102446747427, 19.38674548865680747183403431124, 19.91392786339231233054087919378, 20.79374596431632758662307097970, 21.8690691328774537958997787791, 22.789988567252438705594645383476

Graph of the ZZ-function along the critical line