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Properties

Label 2-405-9.4-c1-0-15
Degree 22
Conductor 405405
Sign −0.173−0.984i-0.173 - 0.984i
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−0.999 + 1.73i)4-s + (−0.5 + 0.866i)5-s + 1.99·10-s + (−2.5 − 4.33i)11-s + (−2 + 3.46i)13-s + (1.99 + 3.46i)16-s − 4·17-s − 5·19-s + (−1 − 1.73i)20-s + (−5 + 8.66i)22-s + (−3 + 5.19i)23-s + (−0.499 − 0.866i)25-s + 7.99·26-s + (2.5 + 4.33i)29-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s + (−0.223 + 0.387i)5-s + 0.632·10-s + (−0.753 − 1.30i)11-s + (−0.554 + 0.960i)13-s + (0.499 + 0.866i)16-s − 0.970·17-s − 1.14·19-s + (−0.223 − 0.387i)20-s + (−1.06 + 1.84i)22-s + (−0.625 + 1.08i)23-s + (−0.0999 − 0.173i)25-s + 1.56·26-s + (0.464 + 0.804i)29-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((−0.173−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((−0.173−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −0.173−0.984i-0.173 - 0.984i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(271,⋅)\chi_{405} (271, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 405, ( :1/2), −0.173−0.984i)(2,\ 405,\ (\ :1/2),\ -0.173 - 0.984i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good2 1+(1+1.73i)T+(−1+1.73i)T2 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2}
7 1+(−3.5+6.06i)T2 1 + (-3.5 + 6.06i)T^{2}
11 1+(2.5+4.33i)T+(−5.5+9.52i)T2 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2}
13 1+(2−3.46i)T+(−6.5−11.2i)T2 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2}
17 1+4T+17T2 1 + 4T + 17T^{2}
19 1+5T+19T2 1 + 5T + 19T^{2}
23 1+(3−5.19i)T+(−11.5−19.9i)T2 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.5−4.33i)T+(−14.5+25.1i)T2 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2}
31 1+(−4.5+7.79i)T+(−15.5−26.8i)T2 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2}
37 1+10T+37T2 1 + 10T + 37T^{2}
41 1+(3.5−6.06i)T+(−20.5−35.5i)T2 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2}
43 1+(−1−1.73i)T+(−21.5+37.2i)T2 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2}
47 1+(1+1.73i)T+(−23.5+40.7i)T2 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2}
53 1−8T+53T2 1 - 8T + 53T^{2}
59 1+(−0.5+0.866i)T+(−29.5−51.0i)T2 1 + (-0.5 + 0.866i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1−1.73i)T+(−30.5+52.8i)T2 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2}
67 1+(3−5.19i)T+(−33.5−58.0i)T2 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2}
71 1−T+71T2 1 - T + 71T^{2}
73 1+8T+73T2 1 + 8T + 73T^{2}
79 1+(6+10.3i)T+(−39.5+68.4i)T2 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2}
83 1+(3+5.19i)T+(−41.5+71.8i)T2 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2}
89 1+9T+89T2 1 + 9T + 89T^{2}
97 1+(7+12.1i)T+(−48.5+84.0i)T2 1 + (7 + 12.1i)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

−10.62294358370808802227290137830, −9.954365582767116809296842721188, −8.840443085737637672864224671104, −8.282841994599585323249472568018, −6.92315502455055926209526901200, −5.82378390722174223213057106595, −4.23024492524889618882193378555, −3.05208159602070366829036127000, −1.98958486613321615527244991942, 0, 2.47450709069665290588655485953, 4.44088180038416228567280094359, 5.34129194771495489156303064358, 6.57561704634863464608148049758, 7.29090938036221534532074082949, 8.265369837306651636680986082671, 8.761550421534312872876249621153, 10.02796411348234986452558275639, 10.52755959451468544356437132672

Graph of the ZZ-function along the critical line