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Properties

Label 2-435-29.7-c1-0-2
Degree 22
Conductor 435435
Sign −0.977+0.209i-0.977 + 0.209i
Analytic cond. 3.473493.47349
Root an. cond. 1.863731.86373
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  + (0.412 + 1.80i)2-s + (−0.900 − 0.433i)3-s + (−1.29 + 0.625i)4-s + (0.222 + 0.974i)5-s + (0.412 − 1.80i)6-s + (−2.78 − 1.34i)7-s + (0.645 + 0.809i)8-s + (0.623 + 0.781i)9-s + (−1.67 + 0.804i)10-s + (−1.04 + 1.31i)11-s + 1.44·12-s + (−4.34 + 5.44i)13-s + (1.27 − 5.60i)14-s + (0.222 − 0.974i)15-s + (−2.99 + 3.75i)16-s − 2.75·17-s + ⋯
L(s)  = 1  + (0.291 + 1.27i)2-s + (−0.520 − 0.250i)3-s + (−0.649 + 0.312i)4-s + (0.0995 + 0.436i)5-s + (0.168 − 0.738i)6-s + (−1.05 − 0.507i)7-s + (0.228 + 0.286i)8-s + (0.207 + 0.260i)9-s + (−0.528 + 0.254i)10-s + (−0.316 + 0.396i)11-s + 0.416·12-s + (−1.20 + 1.51i)13-s + (0.341 − 1.49i)14-s + (0.0574 − 0.251i)15-s + (−0.748 + 0.939i)16-s − 0.667·17-s + ⋯

Functional equation

Λ(s)=(435s/2 ΓC(s) L(s)=((−0.977+0.209i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(435s/2 ΓC(s+1/2) L(s)=((−0.977+0.209i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    3⋅5⋅293 \cdot 5 \cdot 29
Sign: −0.977+0.209i-0.977 + 0.209i
Analytic conductor: 3.473493.47349
Root analytic conductor: 1.863731.86373
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ435(181,⋅)\chi_{435} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 435, ( :1/2), −0.977+0.209i)(2,\ 435,\ (\ :1/2),\ -0.977 + 0.209i)

Particular Values

L(1)L(1) ≈\approx 0.0872512−0.825058i0.0872512 - 0.825058i
L(12)L(\frac12) ≈\approx 0.0872512−0.825058i0.0872512 - 0.825058i
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+(0.900+0.433i)T 1 + (0.900 + 0.433i)T
5 1+(−0.222−0.974i)T 1 + (-0.222 - 0.974i)T
29 1+(−5.26+1.14i)T 1 + (-5.26 + 1.14i)T
good2 1+(−0.412−1.80i)T+(−1.80+0.867i)T2 1 + (-0.412 - 1.80i)T + (-1.80 + 0.867i)T^{2}
7 1+(2.78+1.34i)T+(4.36+5.47i)T2 1 + (2.78 + 1.34i)T + (4.36 + 5.47i)T^{2}
11 1+(1.04−1.31i)T+(−2.44−10.7i)T2 1 + (1.04 - 1.31i)T + (-2.44 - 10.7i)T^{2}
13 1+(4.34−5.44i)T+(−2.89−12.6i)T2 1 + (4.34 - 5.44i)T + (-2.89 - 12.6i)T^{2}
17 1+2.75T+17T2 1 + 2.75T + 17T^{2}
19 1+(−0.370+0.178i)T+(11.8−14.8i)T2 1 + (-0.370 + 0.178i)T + (11.8 - 14.8i)T^{2}
23 1+(−0.230+1.00i)T+(−20.7−9.97i)T2 1 + (-0.230 + 1.00i)T + (-20.7 - 9.97i)T^{2}
31 1+(−1.18−5.18i)T+(−27.9+13.4i)T2 1 + (-1.18 - 5.18i)T + (-27.9 + 13.4i)T^{2}
37 1+(0.609+0.764i)T+(−8.23+36.0i)T2 1 + (0.609 + 0.764i)T + (-8.23 + 36.0i)T^{2}
41 1+2.72T+41T2 1 + 2.72T + 41T^{2}
43 1+(−1.18+5.18i)T+(−38.7−18.6i)T2 1 + (-1.18 + 5.18i)T + (-38.7 - 18.6i)T^{2}
47 1+(−2.28+2.86i)T+(−10.4−45.8i)T2 1 + (-2.28 + 2.86i)T + (-10.4 - 45.8i)T^{2}
53 1+(−2.46−10.7i)T+(−47.7+22.9i)T2 1 + (-2.46 - 10.7i)T + (-47.7 + 22.9i)T^{2}
59 1−0.766T+59T2 1 - 0.766T + 59T^{2}
61 1+(−9.96−4.79i)T+(38.0+47.6i)T2 1 + (-9.96 - 4.79i)T + (38.0 + 47.6i)T^{2}
67 1+(1.18+1.48i)T+(−14.9+65.3i)T2 1 + (1.18 + 1.48i)T + (-14.9 + 65.3i)T^{2}
71 1+(−4.45+5.58i)T+(−15.7−69.2i)T2 1 + (-4.45 + 5.58i)T + (-15.7 - 69.2i)T^{2}
73 1+(3.60−15.8i)T+(−65.7−31.6i)T2 1 + (3.60 - 15.8i)T + (-65.7 - 31.6i)T^{2}
79 1+(−3.51−4.41i)T+(−17.5+77.0i)T2 1 + (-3.51 - 4.41i)T + (-17.5 + 77.0i)T^{2}
83 1+(−8.93+4.30i)T+(51.7−64.8i)T2 1 + (-8.93 + 4.30i)T + (51.7 - 64.8i)T^{2}
89 1+(1.17+5.13i)T+(−80.1+38.6i)T2 1 + (1.17 + 5.13i)T + (-80.1 + 38.6i)T^{2}
97 1+(7.56−3.64i)T+(60.4−75.8i)T2 1 + (7.56 - 3.64i)T + (60.4 - 75.8i)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

−11.73331914381092279911070259041, −10.61770493018053490295794606580, −9.843126653071107032486382074945, −8.728756807108625970508610358301, −7.32805087838512960647138331247, −6.91085756529566966229870800465, −6.34178195720168060059746219442, −5.10974293566735202912674288779, −4.20040379958076696136839576379, −2.34402088588052324053978815715, 0.47823968367375775903247101043, 2.48222550322781498234772879359, 3.34841393730486730376770677846, 4.69274427342329548777880379311, 5.58356391024381565873503554972, 6.73715111821151005185241111576, 8.053218950915987365344504450454, 9.394863567503954433351038970113, 9.968211684467723635757569981115, 10.67776797276682000361960061320

Graph of the ZZ-function along the critical line