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Properties

Label 2-234-117.103-c1-0-11
Degree 22
Conductor 234234
Sign −0.998−0.0499i-0.998 - 0.0499i
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 + (0.180 + 3.60i)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 + (0.0499 + 0.998i)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)=((−0.998−0.0499i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0499i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((−0.998−0.0499i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0499i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −0.998−0.0499i-0.998 - 0.0499i
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), −0.998−0.0499i)(2,\ 234,\ (\ :1/2),\ -0.998 - 0.0499i)

Particular Values

L(1)L(1) ≈\approx 0.00459662+0.184106i0.00459662 + 0.184106i
L(12)L(\frac12) ≈\approx 0.00459662+0.184106i0.00459662 + 0.184106i
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+(−0.180−3.60i)T 1 + (-0.180 - 3.60i)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

−11.62680740226620312902661429905, −11.01780376987087781608488538813, −9.376015341933885465243554856698, −8.597116239560738905760335011207, −7.64391275421923405910319469033, −6.95305833587971595810862731125, −5.72675993555264815172923726517, −4.28576477999555865103491713814, −2.20206854333585498078501237327, −0.17479350460134072723781290997, 3.02627355683493464037854869151, 3.79769189613992710130313349386, 5.30692541755302680743277555490, 6.83421081267066051515075865832, 7.975260229757785515311372105147, 8.788940066070721758208790775687, 10.02620653277006528822256853189, 10.74994293600926345842159691503, 11.29110203307897059270011410484, 12.33231219484803472913442046001

Graph of the ZZ-function along the critical line