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Properties

Label 2-352-44.39-c1-0-8
Degree 22
Conductor 352352
Sign 0.374+0.927i0.374 + 0.927i
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  + (1.68 − 2.31i)3-s + (−0.612 + 1.88i)5-s + (1.05 − 0.767i)7-s + (−1.60 − 4.92i)9-s + (0.361 − 3.29i)11-s + (3.77 − 1.22i)13-s + (3.33 + 4.58i)15-s + (−5.89 − 1.91i)17-s + (4.95 + 3.60i)19-s − 3.73i·21-s + 0.399i·23-s + (0.865 + 0.628i)25-s + (−5.92 − 1.92i)27-s + (−0.0810 − 0.111i)29-s + (−8.27 + 2.69i)31-s + ⋯
L(s)  = 1  + (0.970 − 1.33i)3-s + (−0.273 + 0.843i)5-s + (0.399 − 0.290i)7-s + (−0.533 − 1.64i)9-s + (0.109 − 0.994i)11-s + (1.04 − 0.340i)13-s + (0.860 + 1.18i)15-s + (−1.42 − 0.464i)17-s + (1.13 + 0.826i)19-s − 0.815i·21-s + 0.0833i·23-s + (0.173 + 0.125i)25-s + (−1.14 − 0.370i)27-s + (−0.0150 − 0.0207i)29-s + (−1.48 + 0.483i)31-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=((0.374+0.927i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=((0.374+0.927i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 0.374+0.927i0.374 + 0.927i
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ352(127,⋅)\chi_{352} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 0.374+0.927i)(2,\ 352,\ (\ :1/2),\ 0.374 + 0.927i)

Particular Values

L(1)L(1) ≈\approx 1.48641−1.00263i1.48641 - 1.00263i
L(12)L(\frac12) ≈\approx 1.48641−1.00263i1.48641 - 1.00263i
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+(−0.361+3.29i)T 1 + (-0.361 + 3.29i)T
good3 1+(−1.68+2.31i)T+(−0.927−2.85i)T2 1 + (-1.68 + 2.31i)T + (-0.927 - 2.85i)T^{2}
5 1+(0.612−1.88i)T+(−4.04−2.93i)T2 1 + (0.612 - 1.88i)T + (-4.04 - 2.93i)T^{2}
7 1+(−1.05+0.767i)T+(2.16−6.65i)T2 1 + (-1.05 + 0.767i)T + (2.16 - 6.65i)T^{2}
13 1+(−3.77+1.22i)T+(10.5−7.64i)T2 1 + (-3.77 + 1.22i)T + (10.5 - 7.64i)T^{2}
17 1+(5.89+1.91i)T+(13.7+9.99i)T2 1 + (5.89 + 1.91i)T + (13.7 + 9.99i)T^{2}
19 1+(−4.95−3.60i)T+(5.87+18.0i)T2 1 + (-4.95 - 3.60i)T + (5.87 + 18.0i)T^{2}
23 1−0.399iT−23T2 1 - 0.399iT - 23T^{2}
29 1+(0.0810+0.111i)T+(−8.96+27.5i)T2 1 + (0.0810 + 0.111i)T + (-8.96 + 27.5i)T^{2}
31 1+(8.27−2.69i)T+(25.0−18.2i)T2 1 + (8.27 - 2.69i)T + (25.0 - 18.2i)T^{2}
37 1+(8.18−5.94i)T+(11.4−35.1i)T2 1 + (8.18 - 5.94i)T + (11.4 - 35.1i)T^{2}
41 1+(0.410−0.565i)T+(−12.6−38.9i)T2 1 + (0.410 - 0.565i)T + (-12.6 - 38.9i)T^{2}
43 1−5.83T+43T2 1 - 5.83T + 43T^{2}
47 1+(−4.56+6.28i)T+(−14.5−44.6i)T2 1 + (-4.56 + 6.28i)T + (-14.5 - 44.6i)T^{2}
53 1+(−3.99−12.2i)T+(−42.8+31.1i)T2 1 + (-3.99 - 12.2i)T + (-42.8 + 31.1i)T^{2}
59 1+(−1.53−2.11i)T+(−18.2+56.1i)T2 1 + (-1.53 - 2.11i)T + (-18.2 + 56.1i)T^{2}
61 1+(−6.26−2.03i)T+(49.3+35.8i)T2 1 + (-6.26 - 2.03i)T + (49.3 + 35.8i)T^{2}
67 1−1.32iT−67T2 1 - 1.32iT - 67T^{2}
71 1+(9.26+3.00i)T+(57.4+41.7i)T2 1 + (9.26 + 3.00i)T + (57.4 + 41.7i)T^{2}
73 1+(−1.92−2.65i)T+(−22.5+69.4i)T2 1 + (-1.92 - 2.65i)T + (-22.5 + 69.4i)T^{2}
79 1+(−1.49−4.60i)T+(−63.9+46.4i)T2 1 + (-1.49 - 4.60i)T + (-63.9 + 46.4i)T^{2}
83 1+(−2.12+6.54i)T+(−67.1−48.7i)T2 1 + (-2.12 + 6.54i)T + (-67.1 - 48.7i)T^{2}
89 1+11.9T+89T2 1 + 11.9T + 89T^{2}
97 1+(−1.61−4.96i)T+(−78.4+57.0i)T2 1 + (-1.61 - 4.96i)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

−11.30127668424775360993376870963, −10.65207892978607389646764280802, −9.032124675174325219469993726559, −8.416862168767850870047210888322, −7.43584441387303098825310482110, −6.85937571462708223667626968415, −5.73734917045971802847963871680, −3.70450492321192552348867049169, −2.85136042299985997275459948124, −1.35218177361844790472486595473, 2.11132383619404325593358980191, 3.72726708597064259979784033185, 4.47645281235036207352323172469, 5.33990265887801451872275509709, 7.08450721626288237400835496471, 8.378883014054627316405320231999, 8.946412648442292148917166155194, 9.487713418061466397869869312751, 10.68706173145832306109225977846, 11.41380218330147265540350693302

Graph of the ZZ-function along the critical line