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Properties

Label 2-459-9.7-c1-0-14
Degree 22
Conductor 459459
Sign −0.809+0.586i-0.809 + 0.586i
Analytic cond. 3.665133.66513
Root an. cond. 1.914451.91445
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.974 − 1.68i)2-s + (−0.900 − 1.55i)4-s + (−0.974 − 1.68i)5-s + (0.900 − 1.55i)7-s + 0.389·8-s − 3.80·10-s + (0.194 − 0.337i)11-s + (−0.679 − 1.17i)13-s + (−1.75 − 3.03i)14-s + (2.17 − 3.77i)16-s + 17-s − 7.91·19-s + (−1.75 + 3.03i)20-s + (−0.379 − 0.657i)22-s + (0.905 + 1.56i)23-s + ⋯
L(s)  = 1  + (0.689 − 1.19i)2-s + (−0.450 − 0.779i)4-s + (−0.435 − 0.754i)5-s + (0.340 − 0.589i)7-s + 0.137·8-s − 1.20·10-s + (0.0587 − 0.101i)11-s + (−0.188 − 0.326i)13-s + (−0.468 − 0.812i)14-s + (0.544 − 0.943i)16-s + 0.242·17-s − 1.81·19-s + (−0.392 + 0.679i)20-s + (−0.0809 − 0.140i)22-s + (0.188 + 0.326i)23-s + ⋯

Functional equation

Λ(s)=(459s/2 ΓC(s) L(s)=((−0.809+0.586i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(459s/2 ΓC(s+1/2) L(s)=((−0.809+0.586i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 459459    =    33⋅173^{3} \cdot 17
Sign: −0.809+0.586i-0.809 + 0.586i
Analytic conductor: 3.665133.66513
Root analytic conductor: 1.914451.91445
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ459(154,⋅)\chi_{459} (154, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 459, ( :1/2), −0.809+0.586i)(2,\ 459,\ (\ :1/2),\ -0.809 + 0.586i)

Particular Values

L(1)L(1) ≈\approx 0.584630−1.80432i0.584630 - 1.80432i
L(12)L(\frac12) ≈\approx 0.584630−1.80432i0.584630 - 1.80432i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
17 1−T 1 - T
good2 1+(−0.974+1.68i)T+(−1−1.73i)T2 1 + (-0.974 + 1.68i)T + (-1 - 1.73i)T^{2}
5 1+(0.974+1.68i)T+(−2.5+4.33i)T2 1 + (0.974 + 1.68i)T + (-2.5 + 4.33i)T^{2}
7 1+(−0.900+1.55i)T+(−3.5−6.06i)T2 1 + (-0.900 + 1.55i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.194+0.337i)T+(−5.5−9.52i)T2 1 + (-0.194 + 0.337i)T + (-5.5 - 9.52i)T^{2}
13 1+(0.679+1.17i)T+(−6.5+11.2i)T2 1 + (0.679 + 1.17i)T + (-6.5 + 11.2i)T^{2}
19 1+7.91T+19T2 1 + 7.91T + 19T^{2}
23 1+(−0.905−1.56i)T+(−11.5+19.9i)T2 1 + (-0.905 - 1.56i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.26−3.93i)T+(−14.5−25.1i)T2 1 + (2.26 - 3.93i)T + (-14.5 - 25.1i)T^{2}
31 1+(−2.40−4.15i)T+(−15.5+26.8i)T2 1 + (-2.40 - 4.15i)T + (-15.5 + 26.8i)T^{2}
37 1−7.20T+37T2 1 - 7.20T + 37T^{2}
41 1+(−1.00−1.74i)T+(−20.5+35.5i)T2 1 + (-1.00 - 1.74i)T + (-20.5 + 35.5i)T^{2}
43 1+(−6.28+10.8i)T+(−21.5−37.2i)T2 1 + (-6.28 + 10.8i)T + (-21.5 - 37.2i)T^{2}
47 1+(−2.07+3.59i)T+(−23.5−40.7i)T2 1 + (-2.07 + 3.59i)T + (-23.5 - 40.7i)T^{2}
53 1−2.08T+53T2 1 - 2.08T + 53T^{2}
59 1+(−5.80−10.0i)T+(−29.5+51.0i)T2 1 + (-5.80 - 10.0i)T + (-29.5 + 51.0i)T^{2}
61 1+(4.74−8.21i)T+(−30.5−52.8i)T2 1 + (4.74 - 8.21i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.81−3.14i)T+(−33.5+58.0i)T2 1 + (-1.81 - 3.14i)T + (-33.5 + 58.0i)T^{2}
71 1−14.5T+71T2 1 - 14.5T + 71T^{2}
73 1−0.111T+73T2 1 - 0.111T + 73T^{2}
79 1+(5.36−9.29i)T+(−39.5−68.4i)T2 1 + (5.36 - 9.29i)T + (-39.5 - 68.4i)T^{2}
83 1+(2.10−3.64i)T+(−41.5−71.8i)T2 1 + (2.10 - 3.64i)T + (-41.5 - 71.8i)T^{2}
89 1−2.18T+89T2 1 - 2.18T + 89T^{2}
97 1+(−2.42+4.20i)T+(−48.5−84.0i)T2 1 + (-2.42 + 4.20i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.77584586814750750194941687846, −10.29716179211543987939170338125, −8.981626334136825788498366432233, −8.091088433233491220518188693316, −7.06676361855845962374300929912, −5.52450629528482420843120164346, −4.47413148250163163136288544454, −3.90166903066062545372918324129, −2.48662375311980226648159929474, −1.02622750530232039876315096333, 2.34334423402296457673357168216, 3.94337318674106459856057783989, 4.79516413746234643723416101105, 6.02205610529707992391418579426, 6.59388760321912604152845824681, 7.63411969610798432634401076502, 8.271594479019035312426195912096, 9.477632028753482322005214275376, 10.73750981466086457087899512181, 11.36156186327078599177270630402

Graph of the ZZ-function along the critical line