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Properties

Label 2-234-117.110-c1-0-1
Degree 22
Conductor 234234
Sign 0.744−0.667i0.744 - 0.667i
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 + (−1.53 − 0.810i)3-s + (−0.866 − 0.499i)4-s + (−0.970 + 3.62i)5-s + (−1.17 + 1.26i)6-s + (2.29 + 2.29i)7-s + (−0.707 + 0.707i)8-s + (1.68 + 2.48i)9-s + (3.24 + 1.87i)10-s + (−2.11 − 0.566i)11-s + (0.920 + 1.46i)12-s + (−3.26 − 1.53i)13-s + (2.81 − 1.62i)14-s + (4.42 − 4.75i)15-s + (0.500 + 0.866i)16-s + (3.50 + 6.07i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.883 − 0.467i)3-s + (−0.433 − 0.249i)4-s + (−0.433 + 1.61i)5-s + (−0.481 + 0.517i)6-s + (0.868 + 0.868i)7-s + (−0.249 + 0.249i)8-s + (0.562 + 0.827i)9-s + (1.02 + 0.592i)10-s + (−0.637 − 0.170i)11-s + (0.265 + 0.423i)12-s + (−0.905 − 0.424i)13-s + (0.751 − 0.434i)14-s + (1.14 − 1.22i)15-s + (0.125 + 0.216i)16-s + (0.850 + 1.47i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.744−0.667i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.744−0.667i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.744−0.667i0.744 - 0.667i
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), 0.744−0.667i)(2,\ 234,\ (\ :1/2),\ 0.744 - 0.667i)

Particular Values

L(1)L(1) ≈\approx 0.777370+0.297235i0.777370 + 0.297235i
L(12)L(\frac12) ≈\approx 0.777370+0.297235i0.777370 + 0.297235i
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+(1.53+0.810i)T 1 + (1.53 + 0.810i)T
13 1+(3.26+1.53i)T 1 + (3.26 + 1.53i)T
good5 1+(0.970−3.62i)T+(−4.33−2.5i)T2 1 + (0.970 - 3.62i)T + (-4.33 - 2.5i)T^{2}
7 1+(−2.29−2.29i)T+7iT2 1 + (-2.29 - 2.29i)T + 7iT^{2}
11 1+(2.11+0.566i)T+(9.52+5.5i)T2 1 + (2.11 + 0.566i)T + (9.52 + 5.5i)T^{2}
17 1+(−3.50−6.07i)T+(−8.5+14.7i)T2 1 + (-3.50 - 6.07i)T + (-8.5 + 14.7i)T^{2}
19 1+(−4.70−1.26i)T+(16.4+9.5i)T2 1 + (-4.70 - 1.26i)T + (16.4 + 9.5i)T^{2}
23 1−0.831T+23T2 1 - 0.831T + 23T^{2}
29 1+(8.02−4.63i)T+(14.5−25.1i)T2 1 + (8.02 - 4.63i)T + (14.5 - 25.1i)T^{2}
31 1+(−0.527−0.141i)T+(26.8+15.5i)T2 1 + (-0.527 - 0.141i)T + (26.8 + 15.5i)T^{2}
37 1+(−1.90+0.509i)T+(32.0−18.5i)T2 1 + (-1.90 + 0.509i)T + (32.0 - 18.5i)T^{2}
41 1+(1.24+1.24i)T+41iT2 1 + (1.24 + 1.24i)T + 41iT^{2}
43 1+0.0581iT−43T2 1 + 0.0581iT - 43T^{2}
47 1+(−1.56−5.83i)T+(−40.7+23.5i)T2 1 + (-1.56 - 5.83i)T + (-40.7 + 23.5i)T^{2}
53 1+1.62iT−53T2 1 + 1.62iT - 53T^{2}
59 1+(0.755+2.81i)T+(−51.0+29.5i)T2 1 + (0.755 + 2.81i)T + (-51.0 + 29.5i)T^{2}
61 1−3.57T+61T2 1 - 3.57T + 61T^{2}
67 1+(−9.19+9.19i)T−67iT2 1 + (-9.19 + 9.19i)T - 67iT^{2}
71 1+(−2.31+8.65i)T+(−61.4−35.5i)T2 1 + (-2.31 + 8.65i)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+(5.86−10.1i)T+(−39.5−68.4i)T2 1 + (5.86 - 10.1i)T + (-39.5 - 68.4i)T^{2}
83 1+(−14.0+3.75i)T+(71.8−41.5i)T2 1 + (-14.0 + 3.75i)T + (71.8 - 41.5i)T^{2}
89 1+(3.50+13.0i)T+(−77.0+44.5i)T2 1 + (3.50 + 13.0i)T + (-77.0 + 44.5i)T^{2}
97 1+(−6.97+6.97i)T−97iT2 1 + (-6.97 + 6.97i)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

−12.11512764616403679113706288537, −11.27418521074512839893744123626, −10.74330238637134024113342888274, −9.891155937347618547172116800356, −8.044173267666069847945870484700, −7.37324409416045714094078156030, −5.94876355439971739593482235685, −5.15184951345527431312151920047, −3.37808122360749591924476698695, −2.05789489265320849202748438986, 0.74559027121493917260602139422, 4.06883363584553074411071308856, 5.02209291776911377597755338632, 5.30811500796894446661582709341, 7.26248647746236258255433848750, 7.80798372938560600837692037253, 9.222755105277459584260203440069, 9.912782656054874433429330366167, 11.44733082099355310536843325734, 11.95779340420232499485124719877

Graph of the ZZ-function along the critical line