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Properties

Label 2-352-11.5-c1-0-5
Degree 22
Conductor 352352
Sign 0.998−0.0475i0.998 - 0.0475i
Analytic cond. 2.810732.81073
Root an. cond. 1.676521.67652
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 + 0.363i)3-s + (−0.381 + 1.17i)5-s + (3.61 − 2.62i)7-s + (−0.809 − 2.48i)9-s + (2.19 + 2.48i)11-s + (−1 − 3.07i)13-s + (−0.618 + 0.449i)15-s + (−1.5 + 4.61i)17-s + (6.16 + 4.47i)19-s + 2.76·21-s + 0.763·23-s + (2.80 + 2.04i)25-s + (1.07 − 3.30i)27-s + (−0.381 + 0.277i)29-s + (−2.61 − 8.05i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.209i)3-s + (−0.170 + 0.525i)5-s + (1.36 − 0.993i)7-s + (−0.269 − 0.829i)9-s + (0.660 + 0.750i)11-s + (−0.277 − 0.853i)13-s + (−0.159 + 0.115i)15-s + (−0.363 + 1.11i)17-s + (1.41 + 1.02i)19-s + 0.603·21-s + 0.159·23-s + (0.561 + 0.408i)25-s + (0.206 − 0.635i)27-s + (−0.0709 + 0.0515i)29-s + (−0.470 − 1.44i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.998−0.0475i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.998−0.0475i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.998−0.0475i0.998 - 0.0475i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(225,⋅)\chi_{352} (225, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.998−0.0475i)(2,\ 352,\ (\ :1/2),\ 0.998 - 0.0475i)

Particular Values

L(1)L(1) ≈\approx 1.62402+0.0386167i1.62402 + 0.0386167i
L(12)L(\frac12) ≈\approx 1.62402+0.0386167i1.62402 + 0.0386167i
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
11 1+(−2.19−2.48i)T 1 + (-2.19 - 2.48i)T
good3 1+(−0.5−0.363i)T+(0.927+2.85i)T2 1 + (-0.5 - 0.363i)T + (0.927 + 2.85i)T^{2}
5 1+(0.381−1.17i)T+(−4.04−2.93i)T2 1 + (0.381 - 1.17i)T + (-4.04 - 2.93i)T^{2}
7 1+(−3.61+2.62i)T+(2.16−6.65i)T2 1 + (-3.61 + 2.62i)T + (2.16 - 6.65i)T^{2}
13 1+(1+3.07i)T+(−10.5+7.64i)T2 1 + (1 + 3.07i)T + (-10.5 + 7.64i)T^{2}
17 1+(1.5−4.61i)T+(−13.7−9.99i)T2 1 + (1.5 - 4.61i)T + (-13.7 - 9.99i)T^{2}
19 1+(−6.16−4.47i)T+(5.87+18.0i)T2 1 + (-6.16 - 4.47i)T + (5.87 + 18.0i)T^{2}
23 1−0.763T+23T2 1 - 0.763T + 23T^{2}
29 1+(0.381−0.277i)T+(8.96−27.5i)T2 1 + (0.381 - 0.277i)T + (8.96 - 27.5i)T^{2}
31 1+(2.61+8.05i)T+(−25.0+18.2i)T2 1 + (2.61 + 8.05i)T + (-25.0 + 18.2i)T^{2}
37 1+(3.85−2.80i)T+(11.4−35.1i)T2 1 + (3.85 - 2.80i)T + (11.4 - 35.1i)T^{2}
41 1+(6.73+4.89i)T+(12.6+38.9i)T2 1 + (6.73 + 4.89i)T + (12.6 + 38.9i)T^{2}
43 1+7.09T+43T2 1 + 7.09T + 43T^{2}
47 1+(−2−1.45i)T+(14.5+44.6i)T2 1 + (-2 - 1.45i)T + (14.5 + 44.6i)T^{2}
53 1+(0.763+2.35i)T+(−42.8+31.1i)T2 1 + (0.763 + 2.35i)T + (-42.8 + 31.1i)T^{2}
59 1+(1.69−1.22i)T+(18.2−56.1i)T2 1 + (1.69 - 1.22i)T + (18.2 - 56.1i)T^{2}
61 1+(2−6.15i)T+(−49.3−35.8i)T2 1 + (2 - 6.15i)T + (-49.3 - 35.8i)T^{2}
67 1+3.61T+67T2 1 + 3.61T + 67T^{2}
71 1+(3−9.23i)T+(−57.4−41.7i)T2 1 + (3 - 9.23i)T + (-57.4 - 41.7i)T^{2}
73 1+(5.73−4.16i)T+(22.5−69.4i)T2 1 + (5.73 - 4.16i)T + (22.5 - 69.4i)T^{2}
79 1+(1.85+5.70i)T+(−63.9+46.4i)T2 1 + (1.85 + 5.70i)T + (-63.9 + 46.4i)T^{2}
83 1+(1.42−4.39i)T+(−67.1−48.7i)T2 1 + (1.42 - 4.39i)T + (-67.1 - 48.7i)T^{2}
89 1−8.61T+89T2 1 - 8.61T + 89T^{2}
97 1+(−1.04−3.21i)T+(−78.4+57.0i)T2 1 + (-1.04 - 3.21i)T + (-78.4 + 57.0i)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

−11.47359225357201708189031753941, −10.51021775585035140639259775096, −9.818492292597473859756876599592, −8.581105219113441265979541072088, −7.69152596309609701964949988696, −6.92647126708078113529242683058, −5.54532072691975038745357386987, −4.25270029778607220720638691903, −3.39196203021968693489357201525, −1.49725199424776432196193796858, 1.60178399978884122908616973977, 2.93155851685162470575868904004, 4.88162484982752203680961607332, 5.15774658747839462662738935117, 6.82781049533823022852340650518, 7.86334053196373351174900518745, 8.837347913258328004729102673448, 9.113699778212764753411329746428, 10.83895021703629983305774333074, 11.67621072211849143660359371315

Graph of the ZZ-function along the critical line