↑

Properties

Label 2-234-117.110-c1-0-6
Degree 22
Conductor 234234
Sign 0.955+0.295i0.955 + 0.295i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)2-s + (1.41 + 0.997i)3-s + (−0.866 − 0.499i)4-s + (0.175 − 0.653i)5-s + (1.33 − 1.10i)6-s + (2.85 + 2.85i)7-s + (−0.707 + 0.707i)8-s + (1.00 + 2.82i)9-s + (−0.585 − 0.338i)10-s + (−0.686 − 0.183i)11-s + (−0.726 − 1.57i)12-s + (−1.25 − 3.37i)13-s + (3.49 − 2.01i)14-s + (0.899 − 0.749i)15-s + (0.500 + 0.866i)16-s + (−3.24 − 5.61i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.817 + 0.576i)3-s + (−0.433 − 0.249i)4-s + (0.0782 − 0.292i)5-s + (0.543 − 0.452i)6-s + (1.07 + 1.07i)7-s + (−0.249 + 0.249i)8-s + (0.336 + 0.941i)9-s + (−0.185 − 0.106i)10-s + (−0.206 − 0.0554i)11-s + (−0.209 − 0.453i)12-s + (−0.348 − 0.937i)13-s + (0.934 − 0.539i)14-s + (0.232 − 0.193i)15-s + (0.125 + 0.216i)16-s + (−0.786 − 1.36i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.955+0.295i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.955+0.295i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.955+0.295i0.955 + 0.295i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(227,⋅)\chi_{234} (227, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 0.955+0.295i)(2,\ 234,\ (\ :1/2),\ 0.955 + 0.295i)

Particular Values

L(1)L(1) ≈\approx 1.72339−0.260457i1.72339 - 0.260457i
L(12)L(\frac12) ≈\approx 1.72339−0.260457i1.72339 - 0.260457i
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+(−0.258+0.965i)T 1 + (-0.258 + 0.965i)T
3 1+(−1.41−0.997i)T 1 + (-1.41 - 0.997i)T
13 1+(1.25+3.37i)T 1 + (1.25 + 3.37i)T
good5 1+(−0.175+0.653i)T+(−4.33−2.5i)T2 1 + (-0.175 + 0.653i)T + (-4.33 - 2.5i)T^{2}
7 1+(−2.85−2.85i)T+7iT2 1 + (-2.85 - 2.85i)T + 7iT^{2}
11 1+(0.686+0.183i)T+(9.52+5.5i)T2 1 + (0.686 + 0.183i)T + (9.52 + 5.5i)T^{2}
17 1+(3.24+5.61i)T+(−8.5+14.7i)T2 1 + (3.24 + 5.61i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.253+0.0679i)T+(16.4+9.5i)T2 1 + (0.253 + 0.0679i)T + (16.4 + 9.5i)T^{2}
23 1−1.72T+23T2 1 - 1.72T + 23T^{2}
29 1+(1.11−0.643i)T+(14.5−25.1i)T2 1 + (1.11 - 0.643i)T + (14.5 - 25.1i)T^{2}
31 1+(3.89+1.04i)T+(26.8+15.5i)T2 1 + (3.89 + 1.04i)T + (26.8 + 15.5i)T^{2}
37 1+(7.96−2.13i)T+(32.0−18.5i)T2 1 + (7.96 - 2.13i)T + (32.0 - 18.5i)T^{2}
41 1+(6.55+6.55i)T+41iT2 1 + (6.55 + 6.55i)T + 41iT^{2}
43 1−6.55iT−43T2 1 - 6.55iT - 43T^{2}
47 1+(2.43+9.07i)T+(−40.7+23.5i)T2 1 + (2.43 + 9.07i)T + (-40.7 + 23.5i)T^{2}
53 1−6.34iT−53T2 1 - 6.34iT - 53T^{2}
59 1+(−1.28−4.81i)T+(−51.0+29.5i)T2 1 + (-1.28 - 4.81i)T + (-51.0 + 29.5i)T^{2}
61 1−3.29T+61T2 1 - 3.29T + 61T^{2}
67 1+(7.38−7.38i)T−67iT2 1 + (7.38 - 7.38i)T - 67iT^{2}
71 1+(−2.09+7.82i)T+(−61.4−35.5i)T2 1 + (-2.09 + 7.82i)T + (-61.4 - 35.5i)T^{2}
73 1+(1.40+1.40i)T+73iT2 1 + (1.40 + 1.40i)T + 73iT^{2}
79 1+(−7.29+12.6i)T+(−39.5−68.4i)T2 1 + (-7.29 + 12.6i)T + (-39.5 - 68.4i)T^{2}
83 1+(−3.90+1.04i)T+(71.8−41.5i)T2 1 + (-3.90 + 1.04i)T + (71.8 - 41.5i)T^{2}
89 1+(−2.64−9.85i)T+(−77.0+44.5i)T2 1 + (-2.64 - 9.85i)T + (-77.0 + 44.5i)T^{2}
97 1+(−1.42+1.42i)T−97iT2 1 + (-1.42 + 1.42i)T - 97iT^{2}
show more
show less
   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

−12.06218718195228584119998363970, −11.15214373701501040602168481528, −10.25938295167954976180027701491, −9.052144421243428095029798329268, −8.654265551674834557687883829852, −7.44566624311209679931670124033, −5.28019681649474610472446996121, −4.85386068646394326526537338509, −3.16116900794682104682756825209, −2.07452963107968649640872375981, 1.82762826177729566007012590123, 3.72884354584657292597283471594, 4.76796265976545632686430333166, 6.52758211138307054264966857519, 7.19689901648923734766706353776, 8.142684083484718137700449080358, 8.908622866513041761111967758373, 10.25098527271833662780056202389, 11.26489596956336822168851792694, 12.55236545890629650285981386558

Graph of the ZZ-function along the critical line