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Properties

Label 2-234-117.110-c1-0-3
Degree 22
Conductor 234234
Sign 0.385−0.922i0.385 - 0.922i
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  + (0.258 − 0.965i)2-s + (0.539 + 1.64i)3-s + (−0.866 − 0.499i)4-s + (−0.995 + 3.71i)5-s + (1.72 − 0.0946i)6-s + (−1.54 − 1.54i)7-s + (−0.707 + 0.707i)8-s + (−2.41 + 1.77i)9-s + (3.33 + 1.92i)10-s + (4.04 + 1.08i)11-s + (0.356 − 1.69i)12-s + (−0.109 + 3.60i)13-s + (−1.89 + 1.09i)14-s + (−6.65 + 0.363i)15-s + (0.500 + 0.866i)16-s + (−0.465 − 0.806i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.311 + 0.950i)3-s + (−0.433 − 0.249i)4-s + (−0.445 + 1.66i)5-s + (0.706 − 0.0386i)6-s + (−0.584 − 0.584i)7-s + (−0.249 + 0.249i)8-s + (−0.806 + 0.591i)9-s + (1.05 + 0.607i)10-s + (1.21 + 0.326i)11-s + (0.102 − 0.489i)12-s + (−0.0303 + 0.999i)13-s + (−0.506 + 0.292i)14-s + (−1.71 + 0.0939i)15-s + (0.125 + 0.216i)16-s + (−0.112 − 0.195i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.385−0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.385−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.385−0.922i0.385 - 0.922i
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), 0.385−0.922i)(2,\ 234,\ (\ :1/2),\ 0.385 - 0.922i)

Particular Values

L(1)L(1) ≈\approx 1.00811+0.671208i1.00811 + 0.671208i
L(12)L(\frac12) ≈\approx 1.00811+0.671208i1.00811 + 0.671208i
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+(−0.539−1.64i)T 1 + (-0.539 - 1.64i)T
13 1+(0.109−3.60i)T 1 + (0.109 - 3.60i)T
good5 1+(0.995−3.71i)T+(−4.33−2.5i)T2 1 + (0.995 - 3.71i)T + (-4.33 - 2.5i)T^{2}
7 1+(1.54+1.54i)T+7iT2 1 + (1.54 + 1.54i)T + 7iT^{2}
11 1+(−4.04−1.08i)T+(9.52+5.5i)T2 1 + (-4.04 - 1.08i)T + (9.52 + 5.5i)T^{2}
17 1+(0.465+0.806i)T+(−8.5+14.7i)T2 1 + (0.465 + 0.806i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.86+0.501i)T+(16.4+9.5i)T2 1 + (1.86 + 0.501i)T + (16.4 + 9.5i)T^{2}
23 1−7.82T+23T2 1 - 7.82T + 23T^{2}
29 1+(−6.41+3.70i)T+(14.5−25.1i)T2 1 + (-6.41 + 3.70i)T + (14.5 - 25.1i)T^{2}
31 1+(−1.24−0.333i)T+(26.8+15.5i)T2 1 + (-1.24 - 0.333i)T + (26.8 + 15.5i)T^{2}
37 1+(−4.93+1.32i)T+(32.0−18.5i)T2 1 + (-4.93 + 1.32i)T + (32.0 - 18.5i)T^{2}
41 1+(7.74+7.74i)T+41iT2 1 + (7.74 + 7.74i)T + 41iT^{2}
43 1+1.96iT−43T2 1 + 1.96iT - 43T^{2}
47 1+(−2.52−9.40i)T+(−40.7+23.5i)T2 1 + (-2.52 - 9.40i)T + (-40.7 + 23.5i)T^{2}
53 1−2.04iT−53T2 1 - 2.04iT - 53T^{2}
59 1+(−1.63−6.09i)T+(−51.0+29.5i)T2 1 + (-1.63 - 6.09i)T + (-51.0 + 29.5i)T^{2}
61 1−0.861T+61T2 1 - 0.861T + 61T^{2}
67 1+(−2.36+2.36i)T−67iT2 1 + (-2.36 + 2.36i)T - 67iT^{2}
71 1+(2.90−10.8i)T+(−61.4−35.5i)T2 1 + (2.90 - 10.8i)T + (-61.4 - 35.5i)T^{2}
73 1+(2.45+2.45i)T+73iT2 1 + (2.45 + 2.45i)T + 73iT^{2}
79 1+(−6.38+11.0i)T+(−39.5−68.4i)T2 1 + (-6.38 + 11.0i)T + (-39.5 - 68.4i)T^{2}
83 1+(5.43−1.45i)T+(71.8−41.5i)T2 1 + (5.43 - 1.45i)T + (71.8 - 41.5i)T^{2}
89 1+(0.522+1.95i)T+(−77.0+44.5i)T2 1 + (0.522 + 1.95i)T + (-77.0 + 44.5i)T^{2}
97 1+(12.4−12.4i)T−97iT2 1 + (12.4 - 12.4i)T - 97iT^{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

−11.95489989653346818819974840015, −11.23273612516570377509575258316, −10.51777891432885080601650554615, −9.732406472304928840717189918424, −8.832160158581060440755108684821, −7.17694657667313056487328441107, −6.39827898181157039605878973006, −4.44027798288675956372816251698, −3.66536973299253738227208653432, −2.64979875031814792212423262774, 0.989714828397604233899786077472, 3.30286117911076661182277341688, 4.82473721503391965537226704992, 5.94282307839985057193348239896, 6.91081941951150968857867192771, 8.310415715815148692957243439973, 8.621431244131149750351889451088, 9.519911684566995856999771884730, 11.52741757899511342874451378671, 12.43967501056726781252612764912

Graph of the ZZ-function along the critical line