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Properties

Label 2-234-117.103-c1-0-1
Degree 22
Conductor 234234
Sign −0.835−0.548i-0.835 - 0.548i
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 + (0.523 + 1.65i)3-s + (0.499 − 0.866i)4-s + (0.419 + 0.242i)5-s + (−1.27 − 1.16i)6-s + (−4.37 + 2.52i)7-s + 0.999i·8-s + (−2.45 + 1.72i)9-s − 0.484·10-s + (2.78 − 1.60i)11-s + (1.69 + 0.372i)12-s + (−2.69 + 2.39i)13-s + (2.52 − 4.37i)14-s + (−0.180 + 0.819i)15-s + (−0.5 − 0.866i)16-s + 4.20·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.302 + 0.953i)3-s + (0.249 − 0.433i)4-s + (0.187 + 0.108i)5-s + (−0.521 − 0.476i)6-s + (−1.65 + 0.955i)7-s + 0.353i·8-s + (−0.817 + 0.575i)9-s − 0.153·10-s + (0.838 − 0.484i)11-s + (0.488 + 0.107i)12-s + (−0.748 + 0.663i)13-s + (0.675 − 1.16i)14-s + (−0.0465 + 0.211i)15-s + (−0.125 − 0.216i)16-s + 1.01·17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((−0.835−0.548i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((−0.835−0.548i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −0.835−0.548i-0.835 - 0.548i
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), −0.835−0.548i)(2,\ 234,\ (\ :1/2),\ -0.835 - 0.548i)

Particular Values

L(1)L(1) ≈\approx 0.217488+0.727609i0.217488 + 0.727609i
L(12)L(\frac12) ≈\approx 0.217488+0.727609i0.217488 + 0.727609i
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+(−0.523−1.65i)T 1 + (-0.523 - 1.65i)T
13 1+(2.69−2.39i)T 1 + (2.69 - 2.39i)T
good5 1+(−0.419−0.242i)T+(2.5+4.33i)T2 1 + (-0.419 - 0.242i)T + (2.5 + 4.33i)T^{2}
7 1+(4.37−2.52i)T+(3.5−6.06i)T2 1 + (4.37 - 2.52i)T + (3.5 - 6.06i)T^{2}
11 1+(−2.78+1.60i)T+(5.5−9.52i)T2 1 + (-2.78 + 1.60i)T + (5.5 - 9.52i)T^{2}
17 1−4.20T+17T2 1 - 4.20T + 17T^{2}
19 1−3.21iT−19T2 1 - 3.21iT - 19T^{2}
23 1+(3.13−5.43i)T+(−11.5−19.9i)T2 1 + (3.13 - 5.43i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.29+3.97i)T+(−14.5+25.1i)T2 1 + (2.29 + 3.97i)T + (-14.5 + 25.1i)T^{2}
31 1+(−5.61−3.24i)T+(15.5+26.8i)T2 1 + (-5.61 - 3.24i)T + (15.5 + 26.8i)T^{2}
37 1+2.08iT−37T2 1 + 2.08iT - 37T^{2}
41 1+(−9.57−5.52i)T+(20.5+35.5i)T2 1 + (-9.57 - 5.52i)T + (20.5 + 35.5i)T^{2}
43 1+(−4.73−8.19i)T+(−21.5+37.2i)T2 1 + (-4.73 - 8.19i)T + (-21.5 + 37.2i)T^{2}
47 1+(−4.57+2.64i)T+(23.5−40.7i)T2 1 + (-4.57 + 2.64i)T + (23.5 - 40.7i)T^{2}
53 1−6.41T+53T2 1 - 6.41T + 53T^{2}
59 1+(3.13+1.81i)T+(29.5+51.0i)T2 1 + (3.13 + 1.81i)T + (29.5 + 51.0i)T^{2}
61 1+(−0.500−0.867i)T+(−30.5+52.8i)T2 1 + (-0.500 - 0.867i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.936+0.540i)T+(33.5+58.0i)T2 1 + (0.936 + 0.540i)T + (33.5 + 58.0i)T^{2}
71 1−4.63iT−71T2 1 - 4.63iT - 71T^{2}
73 1+0.325iT−73T2 1 + 0.325iT - 73T^{2}
79 1+(−3.91−6.78i)T+(−39.5+68.4i)T2 1 + (-3.91 - 6.78i)T + (-39.5 + 68.4i)T^{2}
83 1+(−5.08+2.93i)T+(41.5−71.8i)T2 1 + (-5.08 + 2.93i)T + (41.5 - 71.8i)T^{2}
89 1+8.42iT−89T2 1 + 8.42iT - 89T^{2}
97 1+(11.3−6.52i)T+(48.5−84.0i)T2 1 + (11.3 - 6.52i)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

−12.31886349323643690850849151730, −11.57362580706695059872060581876, −10.04993030058563930795884076208, −9.671135368425318772949203821685, −9.007673480722066334742770541184, −7.81606301466223991105650757372, −6.28848727969463874937891352717, −5.70594670386442350599872392075, −3.89186922229373098476020668986, −2.63738366122715544227832534668, 0.71931663584770942631582219792, 2.62164727764586318473373408956, 3.80952257636989411526864995036, 6.01254117880649754959335792000, 7.02599497772931341253132404473, 7.60977773957647379740179390616, 9.060963319071470062423008746497, 9.735832161920703313303338055026, 10.62095862390008449431059041563, 12.12540120672950919225836110084

Graph of the ZZ-function along the critical line