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Properties

Label 2-301-301.102-c1-0-1
Degree 22
Conductor 301301
Sign 0.200−0.979i0.200 - 0.979i
Analytic cond. 2.403492.40349
Root an. cond. 1.550321.55032
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  + (−1.42 + 1.32i)2-s + (−1.61 − 1.50i)3-s + (0.132 − 1.76i)4-s + (−0.0580 + 0.147i)5-s + 4.28·6-s + (−2.61 + 0.374i)7-s + (−0.273 − 0.342i)8-s + (0.139 + 1.86i)9-s + (−0.112 − 0.287i)10-s + (1.73 − 1.18i)11-s + (−2.86 + 2.66i)12-s + (1.98 + 2.48i)13-s + (3.23 − 3.99i)14-s + (0.315 − 0.152i)15-s + (4.35 + 0.655i)16-s + (1.03 + 2.63i)17-s + ⋯
L(s)  = 1  + (−1.00 + 0.934i)2-s + (−0.933 − 0.866i)3-s + (0.0663 − 0.884i)4-s + (−0.0259 + 0.0660i)5-s + 1.74·6-s + (−0.989 + 0.141i)7-s + (−0.0966 − 0.121i)8-s + (0.0465 + 0.621i)9-s + (−0.0356 − 0.0908i)10-s + (0.524 − 0.357i)11-s + (−0.828 + 0.768i)12-s + (0.550 + 0.689i)13-s + (0.864 − 1.06i)14-s + (0.0814 − 0.0392i)15-s + (1.08 + 0.163i)16-s + (0.250 + 0.639i)17-s + ⋯

Functional equation

Λ(s)=(301s/2 ΓC(s) L(s)=((0.200−0.979i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(301s/2 ΓC(s+1/2) L(s)=((0.200−0.979i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 301301    =    7⋅437 \cdot 43
Sign: 0.200−0.979i0.200 - 0.979i
Analytic conductor: 2.403492.40349
Root analytic conductor: 1.550321.55032
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ301(102,⋅)\chi_{301} (102, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 301, ( :1/2), 0.200−0.979i)(2,\ 301,\ (\ :1/2),\ 0.200 - 0.979i)

Particular Values

L(1)L(1) ≈\approx 0.335966+0.274167i0.335966 + 0.274167i
L(12)L(\frac12) ≈\approx 0.335966+0.274167i0.335966 + 0.274167i
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)
bad7 1+(2.61−0.374i)T 1 + (2.61 - 0.374i)T
43 1+(−6.46−1.07i)T 1 + (-6.46 - 1.07i)T
good2 1+(1.42−1.32i)T+(0.149−1.99i)T2 1 + (1.42 - 1.32i)T + (0.149 - 1.99i)T^{2}
3 1+(1.61+1.50i)T+(0.224+2.99i)T2 1 + (1.61 + 1.50i)T + (0.224 + 2.99i)T^{2}
5 1+(0.0580−0.147i)T+(−3.66−3.40i)T2 1 + (0.0580 - 0.147i)T + (-3.66 - 3.40i)T^{2}
11 1+(−1.73+1.18i)T+(4.01−10.2i)T2 1 + (-1.73 + 1.18i)T + (4.01 - 10.2i)T^{2}
13 1+(−1.98−2.48i)T+(−2.89+12.6i)T2 1 + (-1.98 - 2.48i)T + (-2.89 + 12.6i)T^{2}
17 1+(−1.03−2.63i)T+(−12.4+11.5i)T2 1 + (-1.03 - 2.63i)T + (-12.4 + 11.5i)T^{2}
19 1+(0.0683+0.0465i)T+(6.94+17.6i)T2 1 + (0.0683 + 0.0465i)T + (6.94 + 17.6i)T^{2}
23 1+(−0.240−3.20i)T+(−22.7+3.42i)T2 1 + (-0.240 - 3.20i)T + (-22.7 + 3.42i)T^{2}
29 1+(0.0189+0.0829i)T+(−26.1+12.5i)T2 1 + (0.0189 + 0.0829i)T + (-26.1 + 12.5i)T^{2}
31 1+(−2.66−0.822i)T+(25.6+17.4i)T2 1 + (-2.66 - 0.822i)T + (25.6 + 17.4i)T^{2}
37 1+(−5.19−8.99i)T+(−18.5+32.0i)T2 1 + (-5.19 - 8.99i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.703+3.08i)T+(−36.9+17.7i)T2 1 + (0.703 + 3.08i)T + (-36.9 + 17.7i)T^{2}
47 1+(7.22+4.92i)T+(17.1+43.7i)T2 1 + (7.22 + 4.92i)T + (17.1 + 43.7i)T^{2}
53 1+(0.0552+0.140i)T+(−38.8+36.0i)T2 1 + (0.0552 + 0.140i)T + (-38.8 + 36.0i)T^{2}
59 1+(0.525+1.34i)T+(−43.2+40.1i)T2 1 + (0.525 + 1.34i)T + (-43.2 + 40.1i)T^{2}
61 1+(−2.28+0.703i)T+(50.4−34.3i)T2 1 + (-2.28 + 0.703i)T + (50.4 - 34.3i)T^{2}
67 1+(0.341−4.56i)T+(−66.2−9.98i)T2 1 + (0.341 - 4.56i)T + (-66.2 - 9.98i)T^{2}
71 1+(−10.4+5.03i)T+(44.2−55.5i)T2 1 + (-10.4 + 5.03i)T + (44.2 - 55.5i)T^{2}
73 1+(−6.94+1.04i)T+(69.7−21.5i)T2 1 + (-6.94 + 1.04i)T + (69.7 - 21.5i)T^{2}
79 1+(−5.09−8.82i)T+(−39.5+68.4i)T2 1 + (-5.09 - 8.82i)T + (-39.5 + 68.4i)T^{2}
83 1+(−1.89+8.29i)T+(−74.7−36.0i)T2 1 + (-1.89 + 8.29i)T + (-74.7 - 36.0i)T^{2}
89 1+(8.33−2.57i)T+(73.5−50.1i)T2 1 + (8.33 - 2.57i)T + (73.5 - 50.1i)T^{2}
97 1+(10.2+4.91i)T+(60.4+75.8i)T2 1 + (10.2 + 4.91i)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

−11.93654828240678791348532602570, −11.00076234264314949996421961426, −9.777681921844717190499302219742, −9.012597493571933782233114916003, −7.989369933873725363979939269470, −6.78412591650695111301812802390, −6.51745377049782440145654000042, −5.61550050855060679014976658410, −3.54799135258761008847248331069, −1.11441075758458832040517557066, 0.62533048207159575063353192647, 2.78957758519385701137703001022, 4.14112448573167206785050145499, 5.49932576244558282492534880492, 6.52005036730415154252108036010, 8.031468776472691688569188959230, 9.251767817331853482553098936267, 9.787423774829198291495022235762, 10.60562037406433924350171910857, 11.13312900594306710445055920858

Graph of the ZZ-function along the critical line