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Properties

Label 2-234-117.110-c1-0-8
Degree 22
Conductor 234234
Sign 0.646+0.763i0.646 + 0.763i
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 + (−1.15 + 1.29i)3-s + (−0.866 − 0.499i)4-s + (0.817 − 3.04i)5-s + (−0.947 − 1.44i)6-s + (−2.08 − 2.08i)7-s + (0.707 − 0.707i)8-s + (−0.330 − 2.98i)9-s + (2.73 + 1.57i)10-s + (−0.842 − 0.225i)11-s + (1.64 − 0.539i)12-s + (−0.0715 − 3.60i)13-s + (2.54 − 1.47i)14-s + (2.99 + 4.57i)15-s + (0.500 + 0.866i)16-s + (1.80 + 3.11i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.666 + 0.745i)3-s + (−0.433 − 0.249i)4-s + (0.365 − 1.36i)5-s + (−0.386 − 0.591i)6-s + (−0.786 − 0.786i)7-s + (0.249 − 0.249i)8-s + (−0.110 − 0.993i)9-s + (0.864 + 0.499i)10-s + (−0.254 − 0.0680i)11-s + (0.475 − 0.155i)12-s + (−0.0198 − 0.999i)13-s + (0.681 − 0.393i)14-s + (0.772 + 1.18i)15-s + (0.125 + 0.216i)16-s + (0.436 + 0.756i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.646+0.763i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.646+0.763i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.646+0.763i0.646 + 0.763i
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), 0.646+0.763i)(2,\ 234,\ (\ :1/2),\ 0.646 + 0.763i)

Particular Values

L(1)L(1) ≈\approx 0.599057−0.277721i0.599057 - 0.277721i
L(12)L(\frac12) ≈\approx 0.599057−0.277721i0.599057 - 0.277721i
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+(1.15−1.29i)T 1 + (1.15 - 1.29i)T
13 1+(0.0715+3.60i)T 1 + (0.0715 + 3.60i)T
good5 1+(−0.817+3.04i)T+(−4.33−2.5i)T2 1 + (-0.817 + 3.04i)T + (-4.33 - 2.5i)T^{2}
7 1+(2.08+2.08i)T+7iT2 1 + (2.08 + 2.08i)T + 7iT^{2}
11 1+(0.842+0.225i)T+(9.52+5.5i)T2 1 + (0.842 + 0.225i)T + (9.52 + 5.5i)T^{2}
17 1+(−1.80−3.11i)T+(−8.5+14.7i)T2 1 + (-1.80 - 3.11i)T + (-8.5 + 14.7i)T^{2}
19 1+(4.90+1.31i)T+(16.4+9.5i)T2 1 + (4.90 + 1.31i)T + (16.4 + 9.5i)T^{2}
23 1−3.73T+23T2 1 - 3.73T + 23T^{2}
29 1+(−7.44+4.29i)T+(14.5−25.1i)T2 1 + (-7.44 + 4.29i)T + (14.5 - 25.1i)T^{2}
31 1+(4.96+1.33i)T+(26.8+15.5i)T2 1 + (4.96 + 1.33i)T + (26.8 + 15.5i)T^{2}
37 1+(5.77−1.54i)T+(32.0−18.5i)T2 1 + (5.77 - 1.54i)T + (32.0 - 18.5i)T^{2}
41 1+(7.72+7.72i)T+41iT2 1 + (7.72 + 7.72i)T + 41iT^{2}
43 1+6.33iT−43T2 1 + 6.33iT - 43T^{2}
47 1+(−1.63−6.10i)T+(−40.7+23.5i)T2 1 + (-1.63 - 6.10i)T + (-40.7 + 23.5i)T^{2}
53 1−6.73iT−53T2 1 - 6.73iT - 53T^{2}
59 1+(−1.42−5.32i)T+(−51.0+29.5i)T2 1 + (-1.42 - 5.32i)T + (-51.0 + 29.5i)T^{2}
61 1−6.36T+61T2 1 - 6.36T + 61T^{2}
67 1+(−5.85+5.85i)T−67iT2 1 + (-5.85 + 5.85i)T - 67iT^{2}
71 1+(−1.65+6.19i)T+(−61.4−35.5i)T2 1 + (-1.65 + 6.19i)T + (-61.4 - 35.5i)T^{2}
73 1+(−10.6−10.6i)T+73iT2 1 + (-10.6 - 10.6i)T + 73iT^{2}
79 1+(1.30−2.26i)T+(−39.5−68.4i)T2 1 + (1.30 - 2.26i)T + (-39.5 - 68.4i)T^{2}
83 1+(−11.0+2.96i)T+(71.8−41.5i)T2 1 + (-11.0 + 2.96i)T + (71.8 - 41.5i)T^{2}
89 1+(2.10+7.84i)T+(−77.0+44.5i)T2 1 + (2.10 + 7.84i)T + (-77.0 + 44.5i)T^{2}
97 1+(11.9−11.9i)T−97iT2 1 + (11.9 - 11.9i)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

−12.34677667551973034690658898170, −10.64235809335415037471737893923, −10.14144838175629058430742297879, −9.097128861766953144082384357774, −8.294048577081804983738467066579, −6.80337730026939349857853110302, −5.73933516661744511180203917123, −4.92173585474351978694816123815, −3.77965339584959208214003946362, −0.61751188035414591690531273272, 2.10753077327172893559920098240, 3.14533583434654597465246969895, 5.14191632529005620382344376084, 6.49464815641745262480805507677, 6.94568728608342048674683079762, 8.471426248949811056733666615116, 9.695760974095238787466710562327, 10.56098195498160024470776756554, 11.34702760839053301146461651202, 12.20581935246150703566446784062

Graph of the ZZ-function along the critical line