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Properties

Label 2-234-117.16-c1-0-1
Degree 22
Conductor 234234
Sign −0.329−0.944i-0.329 - 0.944i
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  − 2-s + 1.73i·3-s + 4-s + (1.5 + 2.59i)5-s − 1.73i·6-s + (0.5 + 0.866i)7-s − 8-s − 2.99·9-s + (−1.5 − 2.59i)10-s + 1.73i·12-s + (1 − 3.46i)13-s + (−0.5 − 0.866i)14-s + (−4.5 + 2.59i)15-s + 16-s + (−1.5 + 2.59i)17-s + 2.99·18-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.999i·3-s + 0.5·4-s + (0.670 + 1.16i)5-s − 0.707i·6-s + (0.188 + 0.327i)7-s − 0.353·8-s − 0.999·9-s + (−0.474 − 0.821i)10-s + 0.499i·12-s + (0.277 − 0.960i)13-s + (−0.133 − 0.231i)14-s + (−1.16 + 0.670i)15-s + 0.250·16-s + (−0.363 + 0.630i)17-s + 0.707·18-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((−0.329−0.944i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((−0.329−0.944i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −0.329−0.944i-0.329 - 0.944i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(133,⋅)\chi_{234} (133, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), −0.329−0.944i)(2,\ 234,\ (\ :1/2),\ -0.329 - 0.944i)

Particular Values

L(1)L(1) ≈\approx 0.548745+0.773092i0.548745 + 0.773092i
L(12)L(\frac12) ≈\approx 0.548745+0.773092i0.548745 + 0.773092i
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+T 1 + T
3 1−1.73iT 1 - 1.73iT
13 1+(−1+3.46i)T 1 + (-1 + 3.46i)T
good5 1+(−1.5−2.59i)T+(−2.5+4.33i)T2 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2}
7 1+(−0.5−0.866i)T+(−3.5+6.06i)T2 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2}
11 1+11T2 1 + 11T^{2}
17 1+(1.5−2.59i)T+(−8.5−14.7i)T2 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.5+0.866i)T+(−9.5−16.4i)T2 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2}
23 1+(4.5−7.79i)T+(−11.5−19.9i)T2 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+(−0.5−0.866i)T+(−15.5+26.8i)T2 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2}
37 1+(2.5+4.33i)T+(−18.5+32.0i)T2 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2}
41 1+(−4.5+7.79i)T+(−20.5−35.5i)T2 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3.5−6.06i)T+(−21.5+37.2i)T2 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2}
47 1+(−1.5+2.59i)T+(−23.5−40.7i)T2 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1+12T+59T2 1 + 12T + 59T^{2}
61 1+(−3.5−6.06i)T+(−30.5+52.8i)T2 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.5+0.866i)T+(−33.5−58.0i)T2 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2}
71 1+(−7.5+12.9i)T+(−35.5−61.4i)T2 1 + (-7.5 + 12.9i)T + (-35.5 - 61.4i)T^{2}
73 1−14T+73T2 1 - 14T + 73T^{2}
79 1+(2.5−4.33i)T+(−39.5−68.4i)T2 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2}
83 1+(1.5−2.59i)T+(−41.5−71.8i)T2 1 + (1.5 - 2.59i)T + (-41.5 - 71.8i)T^{2}
89 1+(7.5+12.9i)T+(−44.5+77.0i)T2 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2}
97 1+(−6.5−11.2i)T+(−48.5+84.0i)T2 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)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

−12.12212029561317654163808532450, −11.01558892482978496562343132860, −10.48812332318676872274100453231, −9.744585950465248339773846043718, −8.764545666429007937560486500874, −7.67226700054107553368356085374, −6.28518426902667916156014576709, −5.47432360237672440155261835779, −3.61633223728820169754740593841, −2.41514060346324109502670651818, 1.03069248005832328975211427130, 2.31399535242109755378314829303, 4.60937731851655237078049269949, 6.01183268293505075064948135086, 6.88796779699228937821140960342, 8.147201587092163436628687804771, 8.791046144755337286342646609256, 9.719519435748406270377439762616, 10.96907613427425333232802989245, 12.02776659183464191278861516197

Graph of the ZZ-function along the critical line