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Properties

Label 2-234-117.25-c1-0-6
Degree 22
Conductor 234234
Sign 0.365−0.930i0.365 - 0.930i
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 + (−0.215 + 1.71i)3-s + (0.499 + 0.866i)4-s + (2.40 − 1.38i)5-s + (−1.04 + 1.38i)6-s + (0.759 + 0.438i)7-s + 0.999i·8-s + (−2.90 − 0.739i)9-s + 2.77·10-s + (1.92 + 1.11i)11-s + (−1.59 + 0.672i)12-s + (−3.20 − 1.64i)13-s + (0.438 + 0.759i)14-s + (1.86 + 4.43i)15-s + (−0.5 + 0.866i)16-s − 3.80·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.124 + 0.992i)3-s + (0.249 + 0.433i)4-s + (1.07 − 0.621i)5-s + (−0.426 + 0.563i)6-s + (0.286 + 0.165i)7-s + 0.353i·8-s + (−0.969 − 0.246i)9-s + 0.878·10-s + (0.581 + 0.335i)11-s + (−0.460 + 0.194i)12-s + (−0.889 − 0.456i)13-s + (0.117 + 0.202i)14-s + (0.482 + 1.14i)15-s + (−0.125 + 0.216i)16-s − 0.923·17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.365−0.930i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.365−0.930i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.365−0.930i0.365 - 0.930i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(25,⋅)\chi_{234} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 0.365−0.930i)(2,\ 234,\ (\ :1/2),\ 0.365 - 0.930i)

Particular Values

L(1)L(1) ≈\approx 1.52631+1.04093i1.52631 + 1.04093i
L(12)L(\frac12) ≈\approx 1.52631+1.04093i1.52631 + 1.04093i
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+(0.215−1.71i)T 1 + (0.215 - 1.71i)T
13 1+(3.20+1.64i)T 1 + (3.20 + 1.64i)T
good5 1+(−2.40+1.38i)T+(2.5−4.33i)T2 1 + (-2.40 + 1.38i)T + (2.5 - 4.33i)T^{2}
7 1+(−0.759−0.438i)T+(3.5+6.06i)T2 1 + (-0.759 - 0.438i)T + (3.5 + 6.06i)T^{2}
11 1+(−1.92−1.11i)T+(5.5+9.52i)T2 1 + (-1.92 - 1.11i)T + (5.5 + 9.52i)T^{2}
17 1+3.80T+17T2 1 + 3.80T + 17T^{2}
19 1+2.22iT−19T2 1 + 2.22iT - 19T^{2}
23 1+(−0.259−0.449i)T+(−11.5+19.9i)T2 1 + (-0.259 - 0.449i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.81−6.60i)T+(−14.5−25.1i)T2 1 + (3.81 - 6.60i)T + (-14.5 - 25.1i)T^{2}
31 1+(−4.97+2.87i)T+(15.5−26.8i)T2 1 + (-4.97 + 2.87i)T + (15.5 - 26.8i)T^{2}
37 1+11.3iT−37T2 1 + 11.3iT - 37T^{2}
41 1+(−3.52+2.03i)T+(20.5−35.5i)T2 1 + (-3.52 + 2.03i)T + (20.5 - 35.5i)T^{2}
43 1+(−2.81+4.87i)T+(−21.5−37.2i)T2 1 + (-2.81 + 4.87i)T + (-21.5 - 37.2i)T^{2}
47 1+(0.920+0.531i)T+(23.5+40.7i)T2 1 + (0.920 + 0.531i)T + (23.5 + 40.7i)T^{2}
53 1−7.29T+53T2 1 - 7.29T + 53T^{2}
59 1+(3.52−2.03i)T+(29.5−51.0i)T2 1 + (3.52 - 2.03i)T + (29.5 - 51.0i)T^{2}
61 1+(3.94−6.83i)T+(−30.5−52.8i)T2 1 + (3.94 - 6.83i)T + (-30.5 - 52.8i)T^{2}
67 1+(5.95−3.43i)T+(33.5−58.0i)T2 1 + (5.95 - 3.43i)T + (33.5 - 58.0i)T^{2}
71 1−15.9iT−71T2 1 - 15.9iT - 71T^{2}
73 1+5.24iT−73T2 1 + 5.24iT - 73T^{2}
79 1+(7.09−12.2i)T+(−39.5−68.4i)T2 1 + (7.09 - 12.2i)T + (-39.5 - 68.4i)T^{2}
83 1+(−0.641−0.370i)T+(41.5+71.8i)T2 1 + (-0.641 - 0.370i)T + (41.5 + 71.8i)T^{2}
89 1+9.89iT−89T2 1 + 9.89iT - 89T^{2}
97 1+(−13.6−7.86i)T+(48.5+84.0i)T2 1 + (-13.6 - 7.86i)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

−12.46958562878454283180784461315, −11.44630189111850801195664244409, −10.39549564080408204979544257161, −9.345631233391249159283685707441, −8.762254898433017525833121523903, −7.15992475684635156366870668845, −5.81258006090283039034549864465, −5.12377259421575433068951342532, −4.13737105311704984014047091355, −2.40383319136938177395486099518, 1.73249364984201211515764588036, 2.80933503311809719229708038174, 4.67090983846081450916124320396, 6.10288608431691434783171903640, 6.55404338909434768484874293131, 7.79399618388661620345689482052, 9.225120491998427459727987205377, 10.25397358715709947359172304521, 11.30724967352228256760630761007, 11.98419222986432052556116013570

Graph of the ZZ-function along the critical line