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Properties

Label 2-366-61.47-c1-0-0
Degree 22
Conductor 366366
Sign −0.614−0.789i-0.614 - 0.789i
Analytic cond. 2.922522.92252
Root an. cond. 1.709531.70953
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  + (0.5 + 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + (2.19 + 3.79i)5-s + (−0.5 − 0.866i)6-s + (−0.135 − 0.234i)7-s − 0.999·8-s + 9-s + (−2.19 + 3.79i)10-s − 0.270·11-s + (0.499 − 0.866i)12-s + (−1.27 − 2.20i)13-s + (0.135 − 0.234i)14-s + (−2.19 − 3.79i)15-s + (−0.5 − 0.866i)16-s + (−3.42 + 5.92i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s − 0.577·3-s + (−0.249 + 0.433i)4-s + (0.980 + 1.69i)5-s + (−0.204 − 0.353i)6-s + (−0.0511 − 0.0886i)7-s − 0.353·8-s + 0.333·9-s + (−0.693 + 1.20i)10-s − 0.0816·11-s + (0.144 − 0.249i)12-s + (−0.352 − 0.610i)13-s + (0.0361 − 0.0626i)14-s + (−0.566 − 0.980i)15-s + (−0.125 − 0.216i)16-s + (−0.829 + 1.43i)17-s + ⋯

Functional equation

Λ(s)=(366s/2 ΓC(s) L(s)=((−0.614−0.789i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(366s/2 ΓC(s+1/2) L(s)=((−0.614−0.789i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 366366    =    2⋅3⋅612 \cdot 3 \cdot 61
Sign: −0.614−0.789i-0.614 - 0.789i
Analytic conductor: 2.922522.92252
Root analytic conductor: 1.709531.70953
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ366(169,⋅)\chi_{366} (169, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 366, ( :1/2), −0.614−0.789i)(2,\ 366,\ (\ :1/2),\ -0.614 - 0.789i)

Particular Values

L(1)L(1) ≈\approx 0.616926+1.26171i0.616926 + 1.26171i
L(12)L(\frac12) ≈\approx 0.616926+1.26171i0.616926 + 1.26171i
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)
bad2 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
3 1+T 1 + T
61 1+(−7.17−3.09i)T 1 + (-7.17 - 3.09i)T
good5 1+(−2.19−3.79i)T+(−2.5+4.33i)T2 1 + (-2.19 - 3.79i)T + (-2.5 + 4.33i)T^{2}
7 1+(0.135+0.234i)T+(−3.5+6.06i)T2 1 + (0.135 + 0.234i)T + (-3.5 + 6.06i)T^{2}
11 1+0.270T+11T2 1 + 0.270T + 11T^{2}
13 1+(1.27+2.20i)T+(−6.5+11.2i)T2 1 + (1.27 + 2.20i)T + (-6.5 + 11.2i)T^{2}
17 1+(3.42−5.92i)T+(−8.5−14.7i)T2 1 + (3.42 - 5.92i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1+1.73i)T+(−9.5−16.4i)T2 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2}
23 1−5.11T+23T2 1 - 5.11T + 23T^{2}
29 1+(−0.229+0.396i)T+(−14.5−25.1i)T2 1 + (-0.229 + 0.396i)T + (-14.5 - 25.1i)T^{2}
31 1+(−1+1.73i)T+(−15.5−26.8i)T2 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2}
37 1+6.11T+37T2 1 + 6.11T + 37T^{2}
41 1−2.72T+41T2 1 - 2.72T + 41T^{2}
43 1+(−4.82−8.36i)T+(−21.5+37.2i)T2 1 + (-4.82 - 8.36i)T + (-21.5 + 37.2i)T^{2}
47 1+(−4.94+8.56i)T+(−23.5−40.7i)T2 1 + (-4.94 + 8.56i)T + (-23.5 - 40.7i)T^{2}
53 1−8.61T+53T2 1 - 8.61T + 53T^{2}
59 1+(5.38+9.32i)T+(−29.5+51.0i)T2 1 + (5.38 + 9.32i)T + (-29.5 + 51.0i)T^{2}
67 1+(−1.55−2.69i)T+(−33.5+58.0i)T2 1 + (-1.55 - 2.69i)T + (-33.5 + 58.0i)T^{2}
71 1+(−0.729+1.26i)T+(−35.5−61.4i)T2 1 + (-0.729 + 1.26i)T + (-35.5 - 61.4i)T^{2}
73 1+(−2.5+4.33i)T+(−36.5−63.2i)T2 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2}
79 1+(3.84+6.65i)T+(−39.5+68.4i)T2 1 + (3.84 + 6.65i)T + (-39.5 + 68.4i)T^{2}
83 1+(1.86+3.22i)T+(−41.5+71.8i)T2 1 + (1.86 + 3.22i)T + (-41.5 + 71.8i)T^{2}
89 1−13.7T+89T2 1 - 13.7T + 89T^{2}
97 1+(−3.07+5.33i)T+(−48.5−84.0i)T2 1 + (-3.07 + 5.33i)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

−11.59711143097059488862479245753, −10.67656312024268872347107309141, −10.21500934750857306971733249691, −9.014858861002340517890092113886, −7.56783690411986937432772721642, −6.75976730989470201309205422865, −6.12209682195960143587886093672, −5.20062893242564309661969009716, −3.63700910620987928193226476679, −2.37000302787835444948396067627, 0.959667105738758616862177625672, 2.33604205950120538196913605650, 4.36973121353154548142364689998, 5.07283175500246624142591699511, 5.81810472280475581351845273222, 7.12065351943739252519272864986, 8.825684818222730411797383662559, 9.254709291459052360748344957121, 10.14872279371118090352264822472, 11.25645018488591354390139217763

Graph of the ZZ-function along the critical line