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Properties

Label 2-234-117.103-c1-0-3
Degree 22
Conductor 234234
Sign 0.706−0.707i0.706 - 0.707i
Analytic cond. 1.868491.86849
Root an. cond. 1.366931.36693
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  + (0.866 − 0.5i)2-s + (−0.579 + 1.63i)3-s + (0.499 − 0.866i)4-s + (3.32 + 1.91i)5-s + (0.314 + 1.70i)6-s + (−2.91 + 1.68i)7-s − 0.999i·8-s + (−2.32 − 1.89i)9-s + 3.83·10-s + (−1.72 + 0.995i)11-s + (1.12 + 1.31i)12-s + (3.49 − 0.880i)13-s + (−1.68 + 2.91i)14-s + (−5.05 + 4.31i)15-s + (−0.5 − 0.866i)16-s + 2.60·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.334 + 0.942i)3-s + (0.249 − 0.433i)4-s + (1.48 + 0.857i)5-s + (0.128 + 0.695i)6-s + (−1.10 + 0.636i)7-s − 0.353i·8-s + (−0.776 − 0.630i)9-s + 1.21·10-s + (−0.520 + 0.300i)11-s + (0.324 + 0.380i)12-s + (0.969 − 0.244i)13-s + (−0.450 + 0.779i)14-s + (−1.30 + 1.11i)15-s + (−0.125 − 0.216i)16-s + 0.630·17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.706−0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.706−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.706−0.707i0.706 - 0.707i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(103,⋅)\chi_{234} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 0.706−0.707i)(2,\ 234,\ (\ :1/2),\ 0.706 - 0.707i)

Particular Values

L(1)L(1) ≈\approx 1.57429+0.652928i1.57429 + 0.652928i
L(12)L(\frac12) ≈\approx 1.57429+0.652928i1.57429 + 0.652928i
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
3 1+(0.579−1.63i)T 1 + (0.579 - 1.63i)T
13 1+(−3.49+0.880i)T 1 + (-3.49 + 0.880i)T
good5 1+(−3.32−1.91i)T+(2.5+4.33i)T2 1 + (-3.32 - 1.91i)T + (2.5 + 4.33i)T^{2}
7 1+(2.91−1.68i)T+(3.5−6.06i)T2 1 + (2.91 - 1.68i)T + (3.5 - 6.06i)T^{2}
11 1+(1.72−0.995i)T+(5.5−9.52i)T2 1 + (1.72 - 0.995i)T + (5.5 - 9.52i)T^{2}
17 1−2.60T+17T2 1 - 2.60T + 17T^{2}
19 1+1.99iT−19T2 1 + 1.99iT - 19T^{2}
23 1+(−2.13+3.70i)T+(−11.5−19.9i)T2 1 + (-2.13 + 3.70i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.37+7.57i)T+(−14.5+25.1i)T2 1 + (4.37 + 7.57i)T + (-14.5 + 25.1i)T^{2}
31 1+(−4.57−2.64i)T+(15.5+26.8i)T2 1 + (-4.57 - 2.64i)T + (15.5 + 26.8i)T^{2}
37 1−5.08iT−37T2 1 - 5.08iT - 37T^{2}
41 1+(7.13+4.12i)T+(20.5+35.5i)T2 1 + (7.13 + 4.12i)T + (20.5 + 35.5i)T^{2}
43 1+(5.13+8.88i)T+(−21.5+37.2i)T2 1 + (5.13 + 8.88i)T + (-21.5 + 37.2i)T^{2}
47 1+(−4.22+2.44i)T+(23.5−40.7i)T2 1 + (-4.22 + 2.44i)T + (23.5 - 40.7i)T^{2}
53 1+9.16T+53T2 1 + 9.16T + 53T^{2}
59 1+(6.33+3.65i)T+(29.5+51.0i)T2 1 + (6.33 + 3.65i)T + (29.5 + 51.0i)T^{2}
61 1+(−5.63−9.76i)T+(−30.5+52.8i)T2 1 + (-5.63 - 9.76i)T + (-30.5 + 52.8i)T^{2}
67 1+(−4.90−2.83i)T+(33.5+58.0i)T2 1 + (-4.90 - 2.83i)T + (33.5 + 58.0i)T^{2}
71 1−1.94iT−71T2 1 - 1.94iT - 71T^{2}
73 1+8.41iT−73T2 1 + 8.41iT - 73T^{2}
79 1+(−2.97−5.15i)T+(−39.5+68.4i)T2 1 + (-2.97 - 5.15i)T + (-39.5 + 68.4i)T^{2}
83 1+(2.03−1.17i)T+(41.5−71.8i)T2 1 + (2.03 - 1.17i)T + (41.5 - 71.8i)T^{2}
89 1−6.28iT−89T2 1 - 6.28iT - 89T^{2}
97 1+(8.92−5.15i)T+(48.5−84.0i)T2 1 + (8.92 - 5.15i)T + (48.5 - 84.0i)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

−12.32965220508879958296985903473, −11.16603330349314167157988469679, −10.20371519982264707872484203633, −9.904903600551229124411524118908, −8.833792789509025457682747610380, −6.63160030017111797997589659194, −5.99774277640459988646154583221, −5.17684796044451166490612489766, −3.43855660486355485308663168415, −2.53667735561754364405172692476, 1.46678260321413163402632281744, 3.22899845136571553868141625409, 5.13969813407119300144359663660, 5.97464459292821714101500710895, 6.62770271906413540285217847288, 7.902048028967581353837033433382, 9.077153564038840853151108430242, 10.12295325612518342005268557569, 11.26872888490478276080157782076, 12.67743216913891803115338029828

Graph of the ZZ-function along the critical line