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Properties

Label 2-234-117.25-c1-0-12
Degree 22
Conductor 234234
Sign −0.865−0.501i-0.865 - 0.501i
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  + (−0.866 − 0.5i)2-s + (−1.62 − 0.612i)3-s + (0.499 + 0.866i)4-s + (0.548 − 0.316i)5-s + (1.09 + 1.34i)6-s + (−2.15 − 1.24i)7-s − 0.999i·8-s + (2.24 + 1.98i)9-s − 0.633·10-s + (−4.20 − 2.43i)11-s + (−0.279 − 1.70i)12-s + (−0.541 + 3.56i)13-s + (1.24 + 2.15i)14-s + (−1.08 + 0.176i)15-s + (−0.5 + 0.866i)16-s − 6.27·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.935 − 0.353i)3-s + (0.249 + 0.433i)4-s + (0.245 − 0.141i)5-s + (0.447 + 0.547i)6-s + (−0.814 − 0.469i)7-s − 0.353i·8-s + (0.749 + 0.661i)9-s − 0.200·10-s + (−1.26 − 0.732i)11-s + (−0.0806 − 0.493i)12-s + (−0.150 + 0.988i)13-s + (0.332 + 0.575i)14-s + (−0.279 + 0.0456i)15-s + (−0.125 + 0.216i)16-s − 1.52·17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((−0.865−0.501i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((−0.865−0.501i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −0.865−0.501i-0.865 - 0.501i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(25,⋅)\chi_{234} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), −0.865−0.501i)(2,\ 234,\ (\ :1/2),\ -0.865 - 0.501i)

Particular Values

L(1)L(1) ≈\approx 0.0164801+0.0613559i0.0164801 + 0.0613559i
L(12)L(\frac12) ≈\approx 0.0164801+0.0613559i0.0164801 + 0.0613559i
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
3 1+(1.62+0.612i)T 1 + (1.62 + 0.612i)T
13 1+(0.541−3.56i)T 1 + (0.541 - 3.56i)T
good5 1+(−0.548+0.316i)T+(2.5−4.33i)T2 1 + (-0.548 + 0.316i)T + (2.5 - 4.33i)T^{2}
7 1+(2.15+1.24i)T+(3.5+6.06i)T2 1 + (2.15 + 1.24i)T + (3.5 + 6.06i)T^{2}
11 1+(4.20+2.43i)T+(5.5+9.52i)T2 1 + (4.20 + 2.43i)T + (5.5 + 9.52i)T^{2}
17 1+6.27T+17T2 1 + 6.27T + 17T^{2}
19 1−4.86iT−19T2 1 - 4.86iT - 19T^{2}
23 1+(1.77+3.07i)T+(−11.5+19.9i)T2 1 + (1.77 + 3.07i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.415+0.719i)T+(−14.5−25.1i)T2 1 + (-0.415 + 0.719i)T + (-14.5 - 25.1i)T^{2}
31 1+(3.73−2.15i)T+(15.5−26.8i)T2 1 + (3.73 - 2.15i)T + (15.5 - 26.8i)T^{2}
37 1+7.81iT−37T2 1 + 7.81iT - 37T^{2}
41 1+(−0.0678+0.0391i)T+(20.5−35.5i)T2 1 + (-0.0678 + 0.0391i)T + (20.5 - 35.5i)T^{2}
43 1+(−4.84+8.38i)T+(−21.5−37.2i)T2 1 + (-4.84 + 8.38i)T + (-21.5 - 37.2i)T^{2}
47 1+(−4.30−2.48i)T+(23.5+40.7i)T2 1 + (-4.30 - 2.48i)T + (23.5 + 40.7i)T^{2}
53 1+6.36T+53T2 1 + 6.36T + 53T^{2}
59 1+(−7.86+4.54i)T+(29.5−51.0i)T2 1 + (-7.86 + 4.54i)T + (29.5 - 51.0i)T^{2}
61 1+(−5.28+9.15i)T+(−30.5−52.8i)T2 1 + (-5.28 + 9.15i)T + (-30.5 - 52.8i)T^{2}
67 1+(6.82−3.94i)T+(33.5−58.0i)T2 1 + (6.82 - 3.94i)T + (33.5 - 58.0i)T^{2}
71 1−12.3iT−71T2 1 - 12.3iT - 71T^{2}
73 1+1.05iT−73T2 1 + 1.05iT - 73T^{2}
79 1+(1.68−2.92i)T+(−39.5−68.4i)T2 1 + (1.68 - 2.92i)T + (-39.5 - 68.4i)T^{2}
83 1+(13.1+7.60i)T+(41.5+71.8i)T2 1 + (13.1 + 7.60i)T + (41.5 + 71.8i)T^{2}
89 1−0.595iT−89T2 1 - 0.595iT - 89T^{2}
97 1+(−14.7−8.53i)T+(48.5+84.0i)T2 1 + (-14.7 - 8.53i)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

−11.39163348368467541403563190173, −10.70339795454305204597829738759, −9.912805738789492771777381513699, −8.777664799135788031412956359408, −7.52224730974668955739434666437, −6.61106315231499250004758337136, −5.56263424046431513599858521928, −4.01154176665103283429830954797, −2.12506133563451912212272979153, −0.06472313265520329611322090094, 2.60166767609160998247309073477, 4.67918804309072372810674431570, 5.72773206966556483602269880846, 6.61782482537019609845335985198, 7.66549485121842706128334343858, 9.074201843266751410366535498106, 9.941389183062139354725683164570, 10.58487075639055986094949724080, 11.55685884376098180003160271863, 12.76352972109650081417696104440

Graph of the ZZ-function along the critical line