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Properties

Label 2-234-117.103-c1-0-12
Degree 22
Conductor 234234
Sign −0.0806+0.996i-0.0806 + 0.996i
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 + (1.21 − 1.23i)3-s + (0.499 − 0.866i)4-s + (−2.73 − 1.57i)5-s + (0.434 − 1.67i)6-s + (−1.36 + 0.787i)7-s − 0.999i·8-s + (−0.0487 − 2.99i)9-s − 3.15·10-s + (3.26 − 1.88i)11-s + (−0.461 − 1.66i)12-s + (0.899 + 3.49i)13-s + (−0.787 + 1.36i)14-s + (−5.27 + 1.45i)15-s + (−0.5 − 0.866i)16-s + 7.06·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.701 − 0.712i)3-s + (0.249 − 0.433i)4-s + (−1.22 − 0.706i)5-s + (0.177 − 0.684i)6-s + (−0.515 + 0.297i)7-s − 0.353i·8-s + (−0.0162 − 0.999i)9-s − 0.998·10-s + (0.983 − 0.567i)11-s + (−0.133 − 0.481i)12-s + (0.249 + 0.968i)13-s + (−0.210 + 0.364i)14-s + (−1.36 + 0.376i)15-s + (−0.125 − 0.216i)16-s + 1.71·17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((−0.0806+0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0806 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((−0.0806+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0806 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −0.0806+0.996i-0.0806 + 0.996i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(103,⋅)\chi_{234} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), −0.0806+0.996i)(2,\ 234,\ (\ :1/2),\ -0.0806 + 0.996i)

Particular Values

L(1)L(1) ≈\approx 1.18683−1.28670i1.18683 - 1.28670i
L(12)L(\frac12) ≈\approx 1.18683−1.28670i1.18683 - 1.28670i
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+(−1.21+1.23i)T 1 + (-1.21 + 1.23i)T
13 1+(−0.899−3.49i)T 1 + (-0.899 - 3.49i)T
good5 1+(2.73+1.57i)T+(2.5+4.33i)T2 1 + (2.73 + 1.57i)T + (2.5 + 4.33i)T^{2}
7 1+(1.36−0.787i)T+(3.5−6.06i)T2 1 + (1.36 - 0.787i)T + (3.5 - 6.06i)T^{2}
11 1+(−3.26+1.88i)T+(5.5−9.52i)T2 1 + (-3.26 + 1.88i)T + (5.5 - 9.52i)T^{2}
17 1−7.06T+17T2 1 - 7.06T + 17T^{2}
19 1−3.76iT−19T2 1 - 3.76iT - 19T^{2}
23 1+(1.84−3.20i)T+(−11.5−19.9i)T2 1 + (1.84 - 3.20i)T + (-11.5 - 19.9i)T^{2}
29 1+(−0.109−0.189i)T+(−14.5+25.1i)T2 1 + (-0.109 - 0.189i)T + (-14.5 + 25.1i)T^{2}
31 1+(−2.65−1.53i)T+(15.5+26.8i)T2 1 + (-2.65 - 1.53i)T + (15.5 + 26.8i)T^{2}
37 1−0.292iT−37T2 1 - 0.292iT - 37T^{2}
41 1+(6.39+3.69i)T+(20.5+35.5i)T2 1 + (6.39 + 3.69i)T + (20.5 + 35.5i)T^{2}
43 1+(3.05+5.29i)T+(−21.5+37.2i)T2 1 + (3.05 + 5.29i)T + (-21.5 + 37.2i)T^{2}
47 1+(−6.17+3.56i)T+(23.5−40.7i)T2 1 + (-6.17 + 3.56i)T + (23.5 - 40.7i)T^{2}
53 1+14.4T+53T2 1 + 14.4T + 53T^{2}
59 1+(−9.04−5.22i)T+(29.5+51.0i)T2 1 + (-9.04 - 5.22i)T + (29.5 + 51.0i)T^{2}
61 1+(−3.00−5.19i)T+(−30.5+52.8i)T2 1 + (-3.00 - 5.19i)T + (-30.5 + 52.8i)T^{2}
67 1+(6.33+3.65i)T+(33.5+58.0i)T2 1 + (6.33 + 3.65i)T + (33.5 + 58.0i)T^{2}
71 1−0.772iT−71T2 1 - 0.772iT - 71T^{2}
73 1−13.5iT−73T2 1 - 13.5iT - 73T^{2}
79 1+(−6.34−10.9i)T+(−39.5+68.4i)T2 1 + (-6.34 - 10.9i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.314−0.181i)T+(41.5−71.8i)T2 1 + (0.314 - 0.181i)T + (41.5 - 71.8i)T^{2}
89 1+7.06iT−89T2 1 + 7.06iT - 89T^{2}
97 1+(0.535−0.309i)T+(48.5−84.0i)T2 1 + (0.535 - 0.309i)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

−12.02145123408767951084345167380, −11.62300648205128563289799118974, −9.870630339865191437620442281908, −8.858525895562084723612416192837, −8.016544774501680428610182147196, −6.89670210987759316544815688795, −5.74444409714624092867622211688, −3.99428765228656252818037612548, −3.37269240707416043148735516237, −1.34795659634435179667061789313, 3.09207177273390438861760588219, 3.69978281510907938643427472525, 4.82240093081748244250191274566, 6.44788792239047156685714344617, 7.55149038646452515842131844957, 8.192040129866569188071760344769, 9.603458842236166210586608560164, 10.51270363051937595105065219152, 11.53976661376087895129087495304, 12.44627114308290379150557737650

Graph of the ZZ-function along the critical line