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Properties

Label 2-234-117.11-c1-0-6
Degree 22
Conductor 234234
Sign 0.999+0.00318i0.999 + 0.00318i
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  + (0.707 − 0.707i)2-s + (1.04 + 1.38i)3-s − 1.00i·4-s + (1.62 + 0.435i)5-s + (1.71 + 0.239i)6-s + (0.290 + 0.0778i)7-s + (−0.707 − 0.707i)8-s + (−0.823 + 2.88i)9-s + (1.45 − 0.842i)10-s + (−1.12 − 1.12i)11-s + (1.38 − 1.04i)12-s + (−3.56 + 0.535i)13-s + (0.260 − 0.150i)14-s + (1.09 + 2.70i)15-s − 1.00·16-s + (1.26 − 2.19i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.602 + 0.798i)3-s − 0.500i·4-s + (0.727 + 0.194i)5-s + (0.700 + 0.0979i)6-s + (0.109 + 0.0294i)7-s + (−0.250 − 0.250i)8-s + (−0.274 + 0.961i)9-s + (0.461 − 0.266i)10-s + (−0.338 − 0.338i)11-s + (0.399 − 0.301i)12-s + (−0.988 + 0.148i)13-s + (0.0695 − 0.0401i)14-s + (0.282 + 0.698i)15-s − 0.250·16-s + (0.306 − 0.531i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.999+0.00318i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00318i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.999+0.00318i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00318i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.999+0.00318i0.999 + 0.00318i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(11,⋅)\chi_{234} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 0.999+0.00318i)(2,\ 234,\ (\ :1/2),\ 0.999 + 0.00318i)

Particular Values

L(1)L(1) ≈\approx 1.96814−0.00313531i1.96814 - 0.00313531i
L(12)L(\frac12) ≈\approx 1.96814−0.00313531i1.96814 - 0.00313531i
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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
3 1+(−1.04−1.38i)T 1 + (-1.04 - 1.38i)T
13 1+(3.56−0.535i)T 1 + (3.56 - 0.535i)T
good5 1+(−1.62−0.435i)T+(4.33+2.5i)T2 1 + (-1.62 - 0.435i)T + (4.33 + 2.5i)T^{2}
7 1+(−0.290−0.0778i)T+(6.06+3.5i)T2 1 + (-0.290 - 0.0778i)T + (6.06 + 3.5i)T^{2}
11 1+(1.12+1.12i)T+11iT2 1 + (1.12 + 1.12i)T + 11iT^{2}
17 1+(−1.26+2.19i)T+(−8.5−14.7i)T2 1 + (-1.26 + 2.19i)T + (-8.5 - 14.7i)T^{2}
19 1+(−4.16+1.11i)T+(16.4−9.5i)T2 1 + (-4.16 + 1.11i)T + (16.4 - 9.5i)T^{2}
23 1+(−0.660+1.14i)T+(−11.5−19.9i)T2 1 + (-0.660 + 1.14i)T + (-11.5 - 19.9i)T^{2}
29 1+7.48iT−29T2 1 + 7.48iT - 29T^{2}
31 1+(2.78−10.3i)T+(−26.8−15.5i)T2 1 + (2.78 - 10.3i)T + (-26.8 - 15.5i)T^{2}
37 1+(2.37+0.637i)T+(32.0+18.5i)T2 1 + (2.37 + 0.637i)T + (32.0 + 18.5i)T^{2}
41 1+(0.974+3.63i)T+(−35.5+20.5i)T2 1 + (0.974 + 3.63i)T + (-35.5 + 20.5i)T^{2}
43 1+(6.42−3.71i)T+(21.5−37.2i)T2 1 + (6.42 - 3.71i)T + (21.5 - 37.2i)T^{2}
47 1+(−3.46+0.928i)T+(40.7−23.5i)T2 1 + (-3.46 + 0.928i)T + (40.7 - 23.5i)T^{2}
53 1+3.63iT−53T2 1 + 3.63iT - 53T^{2}
59 1+(0.512+0.512i)T+59iT2 1 + (0.512 + 0.512i)T + 59iT^{2}
61 1+(−5.62−9.74i)T+(−30.5+52.8i)T2 1 + (-5.62 - 9.74i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.943+0.252i)T+(58.0−33.5i)T2 1 + (-0.943 + 0.252i)T + (58.0 - 33.5i)T^{2}
71 1+(−1.84−6.87i)T+(−61.4+35.5i)T2 1 + (-1.84 - 6.87i)T + (-61.4 + 35.5i)T^{2}
73 1+(−3.38+3.38i)T−73iT2 1 + (-3.38 + 3.38i)T - 73iT^{2}
79 1+(3.69−6.40i)T+(−39.5−68.4i)T2 1 + (3.69 - 6.40i)T + (-39.5 - 68.4i)T^{2}
83 1+(−3.18−11.8i)T+(−71.8+41.5i)T2 1 + (-3.18 - 11.8i)T + (-71.8 + 41.5i)T^{2}
89 1+(−0.512+1.91i)T+(−77.0−44.5i)T2 1 + (-0.512 + 1.91i)T + (-77.0 - 44.5i)T^{2}
97 1+(−1.61+6.03i)T+(−84.0−48.5i)T2 1 + (-1.61 + 6.03i)T + (-84.0 - 48.5i)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

−12.11926803777674997458921961354, −11.15418307685596137797455191491, −10.05838192456401830355379886850, −9.684760633555167842728248708199, −8.484453771274822053027588808554, −7.14812636656392350548905432744, −5.57199187672814893662138666699, −4.79740141989474723611141602081, −3.32405194246256856200931135975, −2.26923340773432529639157721304, 1.94937650911063441004162488364, 3.37275696966261501684719689553, 5.07510136385105224678174325053, 6.05335940384379495364720069256, 7.26964467323812310588730079212, 7.907770267205417785828651647595, 9.162473658565570833446347925999, 9.985529000715378818356131594002, 11.58330786422956916931153828857, 12.55344430245041206394042988818

Graph of the ZZ-function along the critical line