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Properties

Label 2-3528-7.2-c1-0-30
Degree 22
Conductor 35283528
Sign 0.827+0.561i0.827 + 0.561i
Analytic cond. 28.171228.1712
Root an. cond. 5.307655.30765
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  + (0.707 − 1.22i)5-s − 1.41·13-s + (−0.707 − 1.22i)17-s + (−2 + 3.46i)23-s + (1.50 + 2.59i)25-s + 6·29-s + (2.82 + 4.89i)31-s + (2 − 3.46i)37-s + 7.07·41-s + 4·43-s + (5.65 − 9.79i)47-s + (−2 − 3.46i)53-s + (−2.82 − 4.89i)59-s + (2.12 − 3.67i)61-s + (−1.00 + 1.73i)65-s + ⋯
L(s)  = 1  + (0.316 − 0.547i)5-s − 0.392·13-s + (−0.171 − 0.297i)17-s + (−0.417 + 0.722i)23-s + (0.300 + 0.519i)25-s + 1.11·29-s + (0.508 + 0.879i)31-s + (0.328 − 0.569i)37-s + 1.10·41-s + 0.609·43-s + (0.825 − 1.42i)47-s + (−0.274 − 0.475i)53-s + (−0.368 − 0.637i)59-s + (0.271 − 0.470i)61-s + (−0.124 + 0.214i)65-s + ⋯

Functional equation

Λ(s)=(3528s/2 ΓC(s) L(s)=((0.827+0.561i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3528s/2 ΓC(s+1/2) L(s)=((0.827+0.561i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35283528    =    23⋅32⋅722^{3} \cdot 3^{2} \cdot 7^{2}
Sign: 0.827+0.561i0.827 + 0.561i
Analytic conductor: 28.171228.1712
Root analytic conductor: 5.307655.30765
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3528(3313,⋅)\chi_{3528} (3313, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3528, ( :1/2), 0.827+0.561i)(2,\ 3528,\ (\ :1/2),\ 0.827 + 0.561i)

Particular Values

L(1)L(1) ≈\approx 1.9200043501.920004350
L(12)L(\frac12) ≈\approx 1.9200043501.920004350
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 1
3 1 1
7 1 1
good5 1+(−0.707+1.22i)T+(−2.5−4.33i)T2 1 + (-0.707 + 1.22i)T + (-2.5 - 4.33i)T^{2}
11 1+(−5.5+9.52i)T2 1 + (-5.5 + 9.52i)T^{2}
13 1+1.41T+13T2 1 + 1.41T + 13T^{2}
17 1+(0.707+1.22i)T+(−8.5+14.7i)T2 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2}
19 1+(−9.5−16.4i)T2 1 + (-9.5 - 16.4i)T^{2}
23 1+(2−3.46i)T+(−11.5−19.9i)T2 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+(−2.82−4.89i)T+(−15.5+26.8i)T2 1 + (-2.82 - 4.89i)T + (-15.5 + 26.8i)T^{2}
37 1+(−2+3.46i)T+(−18.5−32.0i)T2 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2}
41 1−7.07T+41T2 1 - 7.07T + 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+(−5.65+9.79i)T+(−23.5−40.7i)T2 1 + (-5.65 + 9.79i)T + (-23.5 - 40.7i)T^{2}
53 1+(2+3.46i)T+(−26.5+45.8i)T2 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2}
59 1+(2.82+4.89i)T+(−29.5+51.0i)T2 1 + (2.82 + 4.89i)T + (-29.5 + 51.0i)T^{2}
61 1+(−2.12+3.67i)T+(−30.5−52.8i)T2 1 + (-2.12 + 3.67i)T + (-30.5 - 52.8i)T^{2}
67 1+(2+3.46i)T+(−33.5+58.0i)T2 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2}
71 1−12T+71T2 1 - 12T + 71T^{2}
73 1+(3.53+6.12i)T+(−36.5+63.2i)T2 1 + (3.53 + 6.12i)T + (-36.5 + 63.2i)T^{2}
79 1+(4−6.92i)T+(−39.5−68.4i)T2 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2}
83 1−5.65T+83T2 1 - 5.65T + 83T^{2}
89 1+(−6.36+11.0i)T+(−44.5−77.0i)T2 1 + (-6.36 + 11.0i)T + (-44.5 - 77.0i)T^{2}
97 1+4.24T+97T2 1 + 4.24T + 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

−8.564499442854433339548822219990, −7.78213480230713943947099081907, −7.04310919169011911185091049256, −6.24424931291055014610273215391, −5.37502384962556578246960770968, −4.81196676248911064419850926058, −3.88185155922298331459972579758, −2.86861767911408905010837352001, −1.88481234374815161419701063678, −0.73763411856540830197876763652, 0.926092722630014605111445645099, 2.36675833024072717194424064416, 2.83885649181238646434063600287, 4.14168364565290695408132210556, 4.67878866564645690286248081441, 5.88976333395735968151038902236, 6.30211889448959292826186510061, 7.13503379597963443294311077008, 7.908428936445971607070469845461, 8.581461037077837911391877267621

Graph of the ZZ-function along the critical line