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Properties

Label 2-294-7.4-c1-0-2
Degree 22
Conductor 294294
Sign 0.386−0.922i0.386 - 0.922i
Analytic cond. 2.347602.34760
Root an. cond. 1.532181.53218
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 + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (1.5 + 2.59i)5-s + 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 + 2.59i)10-s + (−1.5 + 2.59i)11-s + (0.499 + 0.866i)12-s + 4·13-s + 3·15-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (−2 − 3.46i)19-s − 3·20-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.670 + 1.16i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.474 + 0.821i)10-s + (−0.452 + 0.783i)11-s + (0.144 + 0.249i)12-s + 1.10·13-s + 0.774·15-s + (−0.125 − 0.216i)16-s + (0.117 − 0.204i)18-s + (−0.458 − 0.794i)19-s − 0.670·20-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=((0.386−0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+1/2) L(s)=((0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: 0.386−0.922i0.386 - 0.922i
Analytic conductor: 2.347602.34760
Root analytic conductor: 1.532181.53218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ294(67,⋅)\chi_{294} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 294, ( :1/2), 0.386−0.922i)(2,\ 294,\ (\ :1/2),\ 0.386 - 0.922i)

Particular Values

L(1)L(1) ≈\approx 1.47441+0.980753i1.47441 + 0.980753i
L(12)L(\frac12) ≈\approx 1.47441+0.980753i1.47441 + 0.980753i
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1 1
good5 1+(−1.5−2.59i)T+(−2.5+4.33i)T2 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.5−2.59i)T+(−5.5−9.52i)T2 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2}
13 1−4T+13T2 1 - 4T + 13T^{2}
17 1+(−8.5−14.7i)T2 1 + (-8.5 - 14.7i)T^{2}
19 1+(2+3.46i)T+(−9.5+16.4i)T2 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1−9T+29T2 1 - 9T + 29T^{2}
31 1+(0.5−0.866i)T+(−15.5−26.8i)T2 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2}
37 1+(4+6.92i)T+(−18.5+32.0i)T2 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2}
41 1+41T2 1 + 41T^{2}
43 1+10T+43T2 1 + 10T + 43T^{2}
47 1+(3+5.19i)T+(−23.5+40.7i)T2 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.5+2.59i)T+(−26.5−45.8i)T2 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2}
59 1+(−1.5+2.59i)T+(−29.5−51.0i)T2 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2}
61 1+(5+8.66i)T+(−30.5+52.8i)T2 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5+8.66i)T+(−33.5−58.0i)T2 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1+(−1+1.73i)T+(−36.5−63.2i)T2 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2}
79 1+(−0.5−0.866i)T+(−39.5+68.4i)T2 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2}
83 1−9T+83T2 1 - 9T + 83T^{2}
89 1+(−3−5.19i)T+(−44.5+77.0i)T2 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2}
97 1−T+97T2 1 - T + 97T^{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

−12.14814796756962491231152969332, −10.94663304372568654803690152265, −10.12995107750951951828227135541, −8.927862783067826846508157032946, −7.924628609479120144057344214047, −6.77290324066127448857640223373, −6.41161157954552699571805603448, −5.03695989968728934664911248661, −3.44572890759647428492004949300, −2.23391888154599838654479472433, 1.40264919238267836223954523794, 3.08924076079537029196784210471, 4.35612756688499510285107816568, 5.35613875150297397452357254668, 6.24589082920275981337306364604, 8.355759837900032899939204398645, 8.702230901876412996209469040661, 9.882133240093658466959944347700, 10.55647746576914540557831981482, 11.62728483000516633398829236736

Graph of the ZZ-function along the critical line