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Properties

Label 2-234-117.110-c1-0-13
Degree 22
Conductor 234234
Sign −0.943+0.330i-0.943 + 0.330i
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 + (−1.22 − 1.22i)3-s + (−0.866 − 0.499i)4-s + (1.02 − 3.83i)5-s + (−1.50 + 0.862i)6-s + (1.13 + 1.13i)7-s + (−0.707 + 0.707i)8-s + (−0.0126 + 2.99i)9-s + (−3.43 − 1.98i)10-s + (−4.79 − 1.28i)11-s + (0.444 + 1.67i)12-s + (3.50 + 0.826i)13-s + (1.39 − 0.803i)14-s + (−5.95 + 3.42i)15-s + (0.500 + 0.866i)16-s + (−0.584 − 1.01i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.705 − 0.708i)3-s + (−0.433 − 0.249i)4-s + (0.459 − 1.71i)5-s + (−0.613 + 0.352i)6-s + (0.429 + 0.429i)7-s + (−0.249 + 0.249i)8-s + (−0.00423 + 0.999i)9-s + (−1.08 − 0.627i)10-s + (−1.44 − 0.387i)11-s + (0.128 + 0.483i)12-s + (0.973 + 0.229i)13-s + (0.371 − 0.214i)14-s + (−1.53 + 0.883i)15-s + (0.125 + 0.216i)16-s + (−0.141 − 0.245i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((−0.943+0.330i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((−0.943+0.330i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −0.943+0.330i-0.943 + 0.330i
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), −0.943+0.330i)(2,\ 234,\ (\ :1/2),\ -0.943 + 0.330i)

Particular Values

L(1)L(1) ≈\approx 0.171636−1.00970i0.171636 - 1.00970i
L(12)L(\frac12) ≈\approx 0.171636−1.00970i0.171636 - 1.00970i
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+(1.22+1.22i)T 1 + (1.22 + 1.22i)T
13 1+(−3.50−0.826i)T 1 + (-3.50 - 0.826i)T
good5 1+(−1.02+3.83i)T+(−4.33−2.5i)T2 1 + (-1.02 + 3.83i)T + (-4.33 - 2.5i)T^{2}
7 1+(−1.13−1.13i)T+7iT2 1 + (-1.13 - 1.13i)T + 7iT^{2}
11 1+(4.79+1.28i)T+(9.52+5.5i)T2 1 + (4.79 + 1.28i)T + (9.52 + 5.5i)T^{2}
17 1+(0.584+1.01i)T+(−8.5+14.7i)T2 1 + (0.584 + 1.01i)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 1−3.26T+23T2 1 - 3.26T + 23T^{2}
29 1+(−6.50+3.75i)T+(14.5−25.1i)T2 1 + (-6.50 + 3.75i)T + (14.5 - 25.1i)T^{2}
31 1+(−7.25−1.94i)T+(26.8+15.5i)T2 1 + (-7.25 - 1.94i)T + (26.8 + 15.5i)T^{2}
37 1+(4.28−1.14i)T+(32.0−18.5i)T2 1 + (4.28 - 1.14i)T + (32.0 - 18.5i)T^{2}
41 1+(−3.96−3.96i)T+41iT2 1 + (-3.96 - 3.96i)T + 41iT^{2}
43 1−0.889iT−43T2 1 - 0.889iT - 43T^{2}
47 1+(−0.653−2.43i)T+(−40.7+23.5i)T2 1 + (-0.653 - 2.43i)T + (-40.7 + 23.5i)T^{2}
53 1+6.60iT−53T2 1 + 6.60iT - 53T^{2}
59 1+(2.12+7.94i)T+(−51.0+29.5i)T2 1 + (2.12 + 7.94i)T + (-51.0 + 29.5i)T^{2}
61 1−14.1T+61T2 1 - 14.1T + 61T^{2}
67 1+(−2.33+2.33i)T−67iT2 1 + (-2.33 + 2.33i)T - 67iT^{2}
71 1+(0.942−3.51i)T+(−61.4−35.5i)T2 1 + (0.942 - 3.51i)T + (-61.4 - 35.5i)T^{2}
73 1+(3.33+3.33i)T+73iT2 1 + (3.33 + 3.33i)T + 73iT^{2}
79 1+(1.16−2.02i)T+(−39.5−68.4i)T2 1 + (1.16 - 2.02i)T + (-39.5 - 68.4i)T^{2}
83 1+(10.5−2.82i)T+(71.8−41.5i)T2 1 + (10.5 - 2.82i)T + (71.8 - 41.5i)T^{2}
89 1+(−1.78−6.65i)T+(−77.0+44.5i)T2 1 + (-1.78 - 6.65i)T + (-77.0 + 44.5i)T^{2}
97 1+(4.52−4.52i)T−97iT2 1 + (4.52 - 4.52i)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

−11.86436049430502714073475129430, −11.04039771829205781906936585022, −9.999543171211348060917981239790, −8.538636750451539453689116783615, −8.274637000525684924458017045023, −6.29191461329450677152164335018, −5.25864331749796268954659293958, −4.67726967370277872977054707026, −2.30219047117469223151515463721, −0.906112113735969767480484395294, 2.92828853269556760051384227960, 4.29992620083932690138460538484, 5.58628938669139488601429362044, 6.44628196813613205479982037323, 7.32411883871353106782419090098, 8.583885237002034899044042260597, 10.25109982947113481202887633705, 10.45309831808155778528269567741, 11.29036485697885812888988789850, 12.72278892416242944694038509669

Graph of the ZZ-function along the critical line