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Properties

Label 1-6145-6145.1759-r1-0-0
Degree 11
Conductor 61456145
Sign 0.988+0.148i0.988 + 0.148i
Analytic cond. 660.371660.371
Root an. cond. 660.371660.371
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  + (0.576 − 0.817i)2-s + (0.453 + 0.891i)3-s + (−0.336 − 0.941i)4-s + (0.989 + 0.142i)6-s + (−0.383 + 0.923i)7-s + (−0.963 − 0.267i)8-s + (−0.588 + 0.808i)9-s + (−0.984 + 0.178i)11-s + (0.686 − 0.726i)12-s + (0.524 + 0.851i)13-s + (0.533 + 0.845i)14-s + (−0.774 + 0.633i)16-s + (0.777 − 0.629i)17-s + (0.321 + 0.946i)18-s + (0.888 + 0.458i)19-s + ⋯
L(s)  = 1  + (0.576 − 0.817i)2-s + (0.453 + 0.891i)3-s + (−0.336 − 0.941i)4-s + (0.989 + 0.142i)6-s + (−0.383 + 0.923i)7-s + (−0.963 − 0.267i)8-s + (−0.588 + 0.808i)9-s + (−0.984 + 0.178i)11-s + (0.686 − 0.726i)12-s + (0.524 + 0.851i)13-s + (0.533 + 0.845i)14-s + (−0.774 + 0.633i)16-s + (0.777 − 0.629i)17-s + (0.321 + 0.946i)18-s + (0.888 + 0.458i)19-s + ⋯

Functional equation

Λ(s)=(6145s/2 ΓR(s+1) L(s)=((0.988+0.148i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(6145s/2 ΓR(s+1) L(s)=((0.988+0.148i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 61456145    =    5⋅12295 \cdot 1229
Sign: 0.988+0.148i0.988 + 0.148i
Analytic conductor: 660.371660.371
Root analytic conductor: 660.371660.371
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ6145(1759,⋅)\chi_{6145} (1759, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 6145, (1: ), 0.988+0.148i)(1,\ 6145,\ (1:\ ),\ 0.988 + 0.148i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 3.491803763+0.2611597743i3.491803763 + 0.2611597743i
L(12)L(\frac12) ≈\approx 3.491803763+0.2611597743i3.491803763 + 0.2611597743i
L(1)L(1) ≈\approx 1.513567179−0.06955451261i1.513567179 - 0.06955451261i
L(1)L(1) ≈\approx 1.513567179−0.06955451261i1.513567179 - 0.06955451261i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
1229 1 1
good2 1+(0.576−0.817i)T 1 + (0.576 - 0.817i)T
3 1+(0.453+0.891i)T 1 + (0.453 + 0.891i)T
7 1+(−0.383+0.923i)T 1 + (-0.383 + 0.923i)T
11 1+(−0.984+0.178i)T 1 + (-0.984 + 0.178i)T
13 1+(0.524+0.851i)T 1 + (0.524 + 0.851i)T
17 1+(0.777−0.629i)T 1 + (0.777 - 0.629i)T
19 1+(0.888+0.458i)T 1 + (0.888 + 0.458i)T
23 1+(0.243−0.969i)T 1 + (0.243 - 0.969i)T
29 1+(0.656+0.754i)T 1 + (0.656 + 0.754i)T
31 1+(0.621+0.783i)T 1 + (0.621 + 0.783i)T
37 1+(0.796−0.605i)T 1 + (0.796 - 0.605i)T
41 1+(−0.355−0.934i)T 1 + (-0.355 - 0.934i)T
43 1+(0.925+0.379i)T 1 + (0.925 + 0.379i)T
47 1+(0.326−0.945i)T 1 + (0.326 - 0.945i)T
53 1+(−0.555+0.831i)T 1 + (-0.555 + 0.831i)T
59 1+(0.613+0.789i)T 1 + (0.613 + 0.789i)T
61 1+(0.683−0.730i)T 1 + (0.683 - 0.730i)T
67 1+(−0.840−0.542i)T 1 + (-0.840 - 0.542i)T
71 1+(0.898−0.439i)T 1 + (0.898 - 0.439i)T
73 1+(−0.991+0.132i)T 1 + (-0.991 + 0.132i)T
79 1+(−0.267−0.963i)T 1 + (-0.267 - 0.963i)T
83 1+(0.906−0.421i)T 1 + (0.906 - 0.421i)T
89 1+(0.117+0.993i)T 1 + (0.117 + 0.993i)T
97 1+(−0.886−0.462i)T 1 + (-0.886 - 0.462i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−17.59511411345000350571334723771, −16.867609445370880363616410753921, −16.01458875543023748088095974965, −15.54127966707702749992963755786, −14.83549336214401063706656636207, −14.08458866768402754145094084546, −13.44379978390974938360848078117, −13.20200196672967329875490393726, −12.63912767954307818216850136181, −11.745166658361918757895886954096, −11.10161053326623687783530626484, −10.01481113396443512868235430132, −9.483980998186787997434431569677, −8.316071731552313511143085869392, −7.98191902351414040306549250031, −7.49508075462650483341150042982, −6.7926058568866673812248797921, −5.97913769898413155169364004562, −5.586971379889353396413492229700, −4.57721798728135896323682423688, −3.666097232167050139154535631004, −3.10874382961019703311599753595, −2.555595646758603591539715188996, −1.125802287026586210200008124775, −0.585247751016624334541545456565, 0.579773151491958021480667373683, 1.711227009093129630615784458436, 2.60355608052292101477576847774, 2.95179641216642616942838962425, 3.69324458664375821810916693784, 4.54760019479546245149734805138, 5.17226659936405589271350034476, 5.66181979468042415681745401940, 6.50424373702148558312241557173, 7.573227362650748690241522029757, 8.559819616734229650773143898215, 9.03107444465907419555885895546, 9.669514748947941421561667552876, 10.31196415264347429787487604347, 10.81268268679665729496929888752, 11.68878389578920323648250465107, 12.206335707030308466719799090072, 12.86470534754589487200561938029, 13.7803829857308515952392837862, 14.13649892094551130727513002523, 14.82167550511438928478822409253, 15.5249446861259293479599216228, 16.11994113523624285936849715698, 16.42086516292852140973073482294, 17.80541710468464085297414559149

Graph of the ZZ-function along the critical line